# Psycholinguistics/Morphology
## Introduction

Language is an arbitrary method of communication that has many aspects, from speech production, comprehension, and even evolution in terms of the extinction of some words or languages and the creation of new. However, language is also highly structured and can be broken down into smaller units such as phrases, words, morphemes, and phonemes. Since words are just a set of arbitrary symbols, have you ever wondered how words are structured so that we understand their semantic meaning? The topic of this chapter is morphology and it addresses this issue, but what is morphology? Morphology is defined as the study of the internal structure of words (Aronoff, 1983), and discusses one of the above layers of language, morphemes. We will discuss the history and background of morphology, the concept of morphemes, the types of morphemes, morphological theories, the morphological structure of words and the difference in morphology usage and processing in native versus non-native speakers.

## The History of Morphology

Morphology is in some ways both the most ancient and one of the youngest sub-disciplines of linguistics and grammar. It is said to be an ancient study because there is much evidence that the first linguists were mainly morphologists (Haspelmath, 2002). The term morphology is attributed to Johann Wolfgang von Goethe who coined it in the biological context during the nineteenth century. It is for this reason that it is also known as a young sub-discipline (Aronoff, 2005; Haspelmath, 2002). Morphology is defined as the study of the internal structure of words (Aronoff, 1983) but more specifically it is the study of the covariation of the form and meaning of words (Bauer, 2003; Haspelmath, 2002). Although the existence of morphology has been long known, it was overshadowed by phonology and syntax in the early days of generative linguistics. If the internal structure of words was considered at all it was simply placed among syntax or phonology (Spencer and Zwicky, 1998). The era of classic linguistics came to an end with the works of Chomsky who initiated generative grammar, in which a system of rules is capable of generating an infinite number of sentences (Katamba, 1993). For example, in Aspects of Theory of Syntax, Chomsky claimed that certain types of complex words must be treated in a separate or expanded lexicon, effectively separating derivational morphology from syntax (Aronoff, 1983). Due to the influence of Chomsky and others, the discipline of morphology began to flourish and is now a primary aspect in the study of language.

## The Unit of Morphology: The Morpheme
### What is a morpheme?

Language is a multilayered, structural process that can be decomposed into many subunits based on the complexity of the element such as, discourse, phrases, words and phonemes. Morphology is focused on the component of language that is responsible for the formation and semantics of words, the morpheme. A morpheme refers to the smallest meaningful unit of language that a word is made up of. There are many different types of morphemes. Morphemes are described as either being free or bound (Haspelmath, 2002). A free morpheme is a unit that not only has a meaning but is also capable of standing alone; these are often called the “root” of a word. In contrast, although a bound morpheme has a meaning, it must attach to free morphemes; these are often affixes which are found cross-lingually. Furthermore, bound morphemes have additional divisions. The addition of a bound morpheme to a root has two possibilities, it can change the grammar of the word or it can change the semantics of a word (Spencer and Zwicky, 1998). The first, termed inflectional morphology, acts to create a new word-form, in which a word-form is a variant of a word that only changes a minor aspect such as tense, number or possession. For example, the addition of –ed to pass only changes the tense. The latter, termed derivation, acts to create new words or lexemes that have a new semantic meaning. A lexeme is an abstract entity containing sets of word-forms that are stored in the lexicon (Bauer, 2003). An example of derivation is the addition of in− to dependence because it completely changes the semantic meaning. Inflection and derivation will be further discussed later in the chapter.

### Special Morphemes

There are also special cases of morphemes including allomorphs, and unique morphs. An allomorph refers to a morph that realizes a morpheme in a specific context but has another morph in a different context (Katamba, 1993). An example of allomorphy is "A" and "an", in which, “A” is used when the realized morpheme begins with a consonant while “An” is used if the morpheme begins with a vowel (Bauer, 2003). Unique morphs refer to morphemes that only exist in a certain expression of a particular language, for example the allomorph cran− in the English word cranberry (Bauer, 2003). It is important to remember that while special morphemes do exist, inflectional and derivational morphemes are the main types of morphemes and they are further discussed below.

## Inflectional Morphology

Inflection is the aspect of language that is concerned with the relationship between word-forms and lexemes (Haspelmath, 2002), and is a syntactically driven process of word-formation that normally occurs through affixation and does not change the semantic meaning of the base (Katamba, 1993).

### Types of Inflection

Although there are many types of inflection, here however, we will focus on the main division of irregular and regular inflection. Another noteworthy distinction is that of contextual versus inherent inflection. Inherent inflection is not fully determined by syntax even though it does have some syntactic relevance and it deals with what the message is about, i.e. singular vs plural, tense, etc. (Bauer, 2003). Contextual inflection on the other hand, is determined solely by syntactic structure and includes gender, noun class and number (Bauer, 2003).

#### Regular vs. Irregular Inflection

Regular inflection refers to word-forms that can be decomposed into its affixes and bases while the root retains the semantic meaning. Irregular inflection is not capable of this. Irregular inflection refers to the cases of special word-forms that do not retain a root, and therefore cannot be broken down, for example, bought-buy. The distinction between regular and irregular inflection refers primarily to the proposed differences that occur in their processing and storage in the lexicon (Clahsen et al., 2010). Many studies have explored regular and irregular inflection through their priming differences (Marseln-Wilson, 2007). For example, Sonnenstuhl, Eisenbeiss and Clahsen (1999) found that while regular inflection (walked-walk) yields a full priming effect, irregular forms (taught-teach) yield a reduced priming effect; this result was interpreted from the view of dual-mechanism model approach. The dual-mechanism model postulates that regularly inflected words have a morphological representation in the lexicon while irregularly inflected words are stored as whole word-forms that cannot be broken down and are therefore less effective primes; this is a view that is widely held (Clahsen et al., 2010). Therefore, the distinction that exists between regular and irregular inflection is based on the effects on the base and semantics, as well as their differences in processing.

### Inflectional Categories

Inflection has been characterized by four main morphological properties or categories: configurational, agreement, inherent and phrasal (Anderson, 1982).

#### Configurational Properties

Configurational properties help determine which inflection is used through reference to the position and function of the word in a phrase of some other syntactic structure (Katamba, 1993). Configurational properties vary as a function of the class of the word (verb, noun, adjective). For verbs, configurational properties generally refer to the clause that helps determine the relationship of the verb to the noun (Katamba, 1993). While for nouns, configuration is associated with the concept of case (grammatical or oblique [semantic]).

#### Agreement Properties

Agreement Properties refer to characteristics of a word that are determined by the syntactic relations to the other words (Haspelmath, 2002; Katamba, 1993). Types of agreement include NP agreement and noun agreement, in which NP agreement indicates the relationship between noun and person or determiner (Haspelmath, 2002).

#### Inherent Properties

Inherent properties are the properties that can be used according to the agreement rules (Katamba, 1993). Inherent properties include gender, number, and tense and it is these properties that are responsible for determining the words that modify them, such as their adjectives (Haspelmath, 2002).

#### Phrasal Properties

Phrasal Properties are the properties that belong to the whole syntactic phrase but are expressed through a single word (Katamba, 1993). Often, phrasal properties refer to a different type of bound morpheme called a clitic (Spencer and Zwicky, 1998). A clitic can be attached to independent words by syntactic rules. An example is –‘s in the sentence, The Mayor of Lancaster’s limousine, in which it’s the mayor’s limousine but the ‘s is on Lancaster (Katamba, 1993).

## Derivational Morphology

Derivational morphology is the second main branch of morphology and it is concerned with the formation of new lexemes through the addition of bound morphemes. Unlike inflection, derivation acts to change the semantic meaning of a root and/or the class of the word (i.e. noun to verb) (Bauer, 2003; Katamba, 1993).

### The Nature of Derivation

In recent literature there have been three main descriptions of derivation. One account considers derivation to be a simple lexical selection choice (copying an affix on to a word) while another sees derivation as a process or set of morphological operations (Spencer and Zwicky, 1998). The third account of derivation is that of Jackendoff and Bybee, in which derivation is a set of static lexical relations (Spencer and Zwicky, 1998). Advocates of word syntax who reduce the concept of derivation to the selection and application of an affix from the lexicon hold the first account; however this view is dependent upon the existence of an internal hierarchy (Spencer and Zwicky, 1998). The second view, in which derivation is a set of operations, eliminates assumptions about lexical organization by having affixation become a process instead of a selection (Spencer and Zwicky, 1998). The argument of Jackendoff and Bybee is that all derivates have to be listed in the lexicon and that derivational rules simply state “is lexically related” (Spencer and Zwicky, 1998). While these three accounts of derivation are prominent, the actual nature of derivation is still under speculation and subject to scrutiny.

### Types of Derivation

There are four main types of derivation: featural, functional, transposition and expressive. Featural derivation acts on the inherent features of words such as gender, while having no effect on the category of the base (Spencer and Zwicky, 1998). Functional derivation was first distinguished by Kurylowicz who identified that not only did these derivations change the category of the word but also differed semantically from its base, for example recruit→recruiter (Spencer and Zwicky, 1998). Functional derivation is contrasted with transposition, which is a derivation that results in a simple change in category with no affect on function, for example walk→walking (Haspelmath, 2002). The other notable type of derivation is expressive derivation. Expressive derivation does not act on the category or function of a word but instead pertains to the subjective perceptions of the speaker (Spencer and Zwicky, 1998).

### Productive vs. Non-productive Derivation

Morphological rules can be productive or non-productive, in which productive derivations can be further modified to create new words while non-productive derivates cannot, for example, the suffix –al is a non-productive derivate (Haspelmath, 2002). So why would languages have non-productive rules? Simply because they can. Like simple words, non-productive words can merely be listed in the mental lexicon because the lexicon must contain all the information that is not predictable from general rules (Haspelmath, 2002). Just like the differences that exist in inflection processing, there are processing differences between productive and non-productive derivation as well (Clahsen et al., 2010). Through various priming studies, it has been shown that like irregular inflection, non-productive derivation results in a reduced priming and a full whole-word frequency effect, suggesting it is stored in a whole word-form manner in the lexicon (Clahsen et al., 2010). Productive derivation on the other hand, is believed to have a lexical representation through combinatorial entries in which they have a full lexical entry but have a similar internal morphological structure to that of regular inflection, in order to allow for full priming effects (Clahsen et al., 2010). Overall, productive and non-productive derivation differs not only on generative capabilities but in processing and priming effects as well.

## Telling Inflection from Derivation

Numerous criteria have been proposed in order to address how we determine inflection from derivation. It is important to note that many of these criteria have exceptions, and act more as heuristics than laws. We will discuss a few of the criterion and a more comprehensive summary table (Table 1) can be found at the end of this section.
1. Syntactic relevance: While syntax is relevant to inflection, it is not relevant to derivation (Haspelmath, 2002).
2. Obligatoriness: This criterion states that inflectional choices are made from a set of obligatory options imposed by the syntax. In derivation, however, no such obligation exists (Katamba, 1993).
3. Productivity: Productivity refers to the generality of inflection versus derivation. Derivational processes are generally more random while inflectional processes seem to apply more automatically in general (Katamba, 1993).
| Inflection | Derivation |
| 1. Relevant to Syntax | Irrelevant to Syntax |
| 2. Obligatory | Optional |
| 3. Not replaceable by a simple word | replaceable by a simple word |
| 4. Same concept as base | New concept |
| 5. Abstract meaning | Concrete meaning |
| 6. Semantic regularity | Possible semantic irregularity |
| 7. Less relevant to base meaning | very relevant to base meaning |
| 8. Unlimited applicability | Limited applicability |
| 9. Expression at word periphery | Expression near base |
| 10. Less base allomorphy | more base allomorphy |
| 11. Cumulative expression possible | no cumulative expression |
| 12. not repeatable | repeatable |
Table 1: A list of properties of inflection and derivation that helps them be distinguished from one another (Haspelmath, 2002).

## The Past-Tense Debate

One area of constant controversy in morphology is how the past tense of English verbs are mentally represented and processed. The focus of the debate is between the classic views of mental computation, a rule based system or a parallel distributed processing model (connectionist model) (Marslen-Wilson and Tyler, 1998). More recently, Pinker and colleagues proposed a dual-mechanism model for how the past-tense of inflected verbs are processed (McClelland and Patterson, 2002). English inflectional verb morphology contains both regularly and irregularly inflected verbs, with the majority being regularly inflected and only 160 verbs being irregularly inflected (Tyler et al., 2002). Although it is widely accepted that irregular forms are stored as wholes, the mechanism of regular inflection is less certain (Tyler et al., 2002). Some believe that regular inflection involves the same mechanism as irregular inflection, a view held by connectionists such as McClelland et al. (Joanisse and Seidenberg, 2005). Others believe that two separate mechanisms exists, in which, regular inflection involves combinatorial rules while irregular inflection involves whole-word storage in the lexicon (Joanisse and Seidenberg, 2005). Both approaches are supported by behavioral, neuropsychological and neuroimaging evidence, however the evidence has been controversial and inconclusive.

### Dual-Mechanism Approach

Pinker and colleagues proposed a hybrid model called the dual-mechanism approach that consisted of a rule-mechanism for regular inflection and an “associative net” for the exceptions (irregular inflection) (Joanisse and Seidenberg, 2005). The associative net is required because irregular inflection is unpredictable and therefore it is unlikely that irregular inflection is acquired through rules (Marslen-Wilson and Tyler, 1998). A rule is said to be acquired suddenly and uniform in its application. It is also supposed to be independent of the frequency, meaning and phonology of the verb to which it applies (McClelland and Patterson, 2002).. Therefore three key aspects of the dual-mechanism theory are that the acquisition is sudden, application is uniform and independent of other factors, and that the mechanism that is involved in regular inflection differs from that of exceptions. Although this theory has undergone revisions these concepts have continued to be emphasized (McClelland and Patterson, 2002).

### Connectionist Approach

The connectionist approach was proposed by Rumelhart and McClelland and consists of an associative network, in which, relationships are learned between phonology and past-tense forms in English. It basically works as a neural network in which the connection weights are adjusted by occurrences of correct and incorrect experiences (McClelland and Patterson, 2002). The connectionist model states that the regular and irregular inflection operates by a single mechanism and that the acquisition of language is more gradual than the dual-mechanism model proposes (McClelland and Patterson, 2002). Overall, the two prominent theories in the past-tense debate make opposite predictions from one another.

### What does the evidence say
#### Rebuttal

Issues have arisen in the data that initially were used to argue and support the dual-mechanism model (Haskell et al., 2003). This has led connectionists to challenge much of the initial basis of their rivals. It was based on the data of Marcus et al. that acquisition of language was sudden, however, when Hoeffiner evaluated this data he found a more gradual acquisition, and when he did is own analysis it was found to be even more gradual (McClelland & Patterson, 2002). Another key aspect of the dual-mechanism approach was that application is uniform and independent of other characteristics but it has since been found that application is sensitive to phonology (McClelland and Patterson, 2002). The third primary aspect of the dual-mechanism model was the idea of two separate mechanisms. This aspect will be further examined in the evidence below.

#### Correlating Neuropsychological and Neuroimaging Evidence with Behavioral Effects

Much of the evidence that exists in the past-tense debate is based on the comparison of behavioral priming effects in aphasics and correlating the behavioral effect to the particular brain area of the person’s deficit. If the dual-mechanism model is correct in the statement that there are two separate mechanisms for regular and irregular inflection than a distinction should be made evident by the presence or absence of dissociations, in which no dissociation based on morphology should exist if only one mechanism is involved (Marslen-Wilson & Tyler, 1998). Tyler et al. have examined priming effects in both non-fluent aphasics (Frontal lobe/Broca’s Area damage) and semantically impaired patients (temporal lobe damage) in multiple studies (Marslen-Wilson and Tyler, 1998; Tyler et al., 2002). In all cases Tyler et al. found the same pattern, in which, non-fluent aphasics had no significant priming of regular inflections (walk→walked) but had significant priming of both irregular (take→took) and semantic (cello→violin) pairs, while semantically impaired patients demonstrated the opposite pattern (Tyler et al., 2002).
While damage to the frontal lobe impairs regular inflectional priming, and damage to the temporal lobe impairs irregular inflectional priming, is the deficit morphological in nature or due to some other factor? Multiple studies have shown that the dissociations are more likely to be phonological in nature and not morphological. Joanisse and Seidenberg (2005) demonstrated that the dissociation is more likely to be phonological through fMRI and the use of regulars, non-words, pseudoregulars and true irregulars as primes. The concept was that if the dissociation seen in the frontal lobe (inferior frontal gyrus specifically) was due to the morphological properties than both types of irregulars (true irregulars and pseudoregulars) should contrast equally with regular words and non-words. Their results showed that this was not the case and that although true irregulars contrasted with regulars and non-words, pseudoregulars did not. This suggested that the dissociation was due to phonological properties instead of morphological properties (Joanisse and Seidenberg, 2005). Further support for a phonological dissociation comes from research demonstrating that irregular pairs are more similar to regular pairs than semantic pairs (Tyler et al., 2002). Event-related potentials have demonstrated that both regular and irregular pairs elicit a left anterior negativity (LAN) while semantic pairs do not (Marslen-Wilson et al., 2000). It is also known from previous research that purely semantic primes do not last over long delays. However, it has been shown that irregular priming does withstand long delays just as regular priming does. This makes the account that the dissociation often observed is the result of phonological properties, not morphological ones, more plausible. Regular verbs require phonological parsing of their internal structure while irregular verbs are stored as wholes and no such parsing is required (Tyler et al., 2002). One common characteristic of non-fluent aphasics is difficulty in phonological assembly and disassembly (Tyler et. al., 2002). It is these difficulties in phonological parsing that are believed to account for the differential dissociations.
Overall, the evidence in existence for either side of the debate is highly controversial, and the results of neuroimaging studies are inconclusive. While some evidence points to the fact that there is little significant difference in brain activation for regular and irregular priming, it cannot be concluded that the same mechanism leads to the activation in each case (Joanisse & Seidenberg, 2005).

## Morphology Usage in Native vs. Non-Native Speakers

There are many distinctions between first language (L1) and second language (L2) speakers, such as processing being slower in L2 speakers and the absence of the left anterior negativity and delay of other ERP components in L2 speakers (Mueller, 2005). Ullman proposed that processing one’s native language involves two different memory systems: lexical store for memorized words (declarative memory) and a mental grammar (procedural memory) that includes the combinatorial rules. Ullman also proposes that maturational changes lead to the attenuation of the procedural memory and increased use of the declarative (Ullman, 2005). This predicts that in late L2 leaners, morphological decomposition is underused so that most processing relies on lexical storage.

### Native Language Processing
#### Regular and Irregular Inflectional Processing

There are many experimental studies that have led to consistent results on how regular and irregular inflection is processed in native speakers. For example, consistent priming differences between regular and irregular inflection have been found in different languages (Marslen-Wilson, 2007). A study by Sonnenstuhl, Eisenbeiss and Clahsen (1999) found that regular priming pairs yielded a full priming effect while irregular priming pairs only yielded a partial effect (Clahsen et al., 2010). These results were interpreted under the dual-mechanism model where regular primes are morphologically decomposed into a root and affixes, which yielded a full prime, while irregular forms are stored as wholes and cannot be broken down (Clahsen et al., 2010).

#### Derivational Processing

Priming studies as well as word-frequency studies have been used to examine derivational morphology processing in native speakers. Results have shown that non-productive derivation parallels the pattern of irregular inflection, demonstrating a reduced priming effect and whole-word frequency effects (Clahsen et al., 2010). Productive derivation on the other hand produces a full priming effect similar to regular inflection but also demonstrates a whole word frequency effect similar to irregular inflection (Clahsen et al., 2010). This pattern is the result of the fact that productive derivation has a lexical entry (its stem) but is also broken down based on their internal morphological structure. This indicates that a considerable proportion of derived words are stored in the mental lexicon (Marslen-Wilson, 2007).

### Non-native Language Processing
#### Regular and Irregular Inflectional Processing

The results for morphological processing in second language speakers have been controversial, with some research claiming they found differences while other did not (Clahsen et al., 2010). Neubauer and Clahsen (2009) found that in L2 speakers, the frequency of a word has an effect on the lexical decision time for both regular and irregular words but that this is only true for the irregular words in native speakers (Neubauer and Clahsen, 2009). This suggests that L2 processing relies more heavily on lexical memory than L1 processing. Neubauer and Clahsen also performed a priming experiment that demonstrated that both L1 and L2 speakers experience a partial prime for irregularly inflected words, while only L1 experienced a full priming effect with regularly inflected words (Clahsen et al., 2010).

#### Derivational Processing

Derivational morphology in non-native speakers has only been examined in a few studies. One area of focus is whether transfer exists from L1 to L2. Koda (2000) tested Chinese and Korean L2 learners of English in a judgment task that required them to decide if a word could be broken down into further meaningful units. Koda was interested in whether one’s native language has an influence on L2 processing. In terms of Koda's experiment, if L1 does have an effect on L2 processing, Koreans should be better at the task due to more derivation in their language. (Clahsen et al., 2010) However, the results showed no evidence that L1 affects L2 processing (Koda, 2000). Current evidence does suggest that effects of morphological structure seen in L1 are weaker in L2, in which, stem-priming effects are reduced in L2. This is consistent with Ullman’s idea that L2 processing is less affected by their combinatorial structure than L1 processing (Ullman, 2005).
Overall, there are some distinctions in morphological processing of both inflectional and derivational nature between native and non-native speakers. Clearly, further research is required in order to confirm certain findings, especially in the area of derivation.

## Exercises
### Questions

1. Which of the following words are morphologically complex, i.e contains 2 of more morphemes. For each complex word come up with 2 semantically related words.
Fights, independent, bubble, researched, rewrite, staying, stay
2. Is each of the following words derived or inflected? What is the root?
* Independent
* Walked
* Bought
* Formal
3. Identify the words with prefixes by underlying the prefix.
    Unable, independent, receive, establish, uncertain
4. Decompose the following words into morphemes and identify the root in each case.
    Researcher, discouraged, kingdom, unfaithful, believable, truth
5. Divide the following words into the smallest meaningful units and describe the morphological processes of how the original word is form (i.e. Inflection or derivation)
    Unhappiness, discourages, describing
6. The following words have been separated into its component morphemes. Identify each morpheme as bound or free.
* Readings
* Encouragement
* Horses
* Independently
* Jumps
7. Some of the following words contain suffixes, identify them by underlining the suffix.
    Happiness, unkind, roses, judgement
8. Which 5 words from the list below are inflectional? Which are derivational?
| Elements | Within | Independent | Uninhibited |
| Drink | Ordering | Linked | Cram |
| Cat | Taught | Such | Off |
| Unkind | The | Morphology | Jumped |
9. Classify the affixes by their function (i.e. what the force the morpheme to become, such as noun, verb, adjective)
* Reading
* Joyful
* Creative
* Dependent
* Destiny

### Quiz

Fill in the Blanks
1. A ___________________is a unit of language that has a meaning and is capable of standing by itself, while a _________is unit that must be joined to another unit.
2. Teach and Taught demonstrate _______________inflection.
3. The dual-mechanism model states that _____________inflection is stored as whole word-forms while _____________inflection has a morphological representation in the lexicon.
4. _______________ results in the formation of new lexemes
5. ______________derivation changes both the category and semantics of a word while ___________ only changes the category.
Multiple choice
1. Which of the following are not ways to tell inflection from derivation?
a) Syntactic relevance and semantic regularly
b) Obligatoriness and allomorphy
c) Type of meaning and agreement
d) Applicability and concept
2. The past-tense debate refers to how verbs are mentally represented and processed. What are the two main theories?
a) Connectionist model and Dual-mechanism model
b) Connectionist model and rule based model
c) Rule based model and dual-mechanism model
3. What is does the associative net refer to?
a) It refers to the concept of the connectionist model
b) It refers to the mechanism used for processing irregular inflections
c) It refers to the mechanism used for processing regular inflections
4. Event related potentials show that both regular and irregular pairs in priming tasks elict a ______ while semantic pairs do not.
a) N400
b) P100
c) LAN
5. Who experiences only partial priming effects with regularly inflected words?
a) Non-native speakers
b) Native-speakers
c) Infants
d) Youth
True or False→if false, correct the statement
1. Morphology is the area of linguisitics focused on the formation and sound of a word.
2. A word-form is a variant of a word that changes a major aspect.
3. Inflection is concerned with the relationship between word-forms and lexemes
4. The four properties of inflection are: configurational, agreement, inherent and transposition.
5. Productive derivation is thought to have both a full lexical entry (like irregular inflection) and an internal morphological structure similar to regular inflection.
6. Everyone is in agreement on how verbs are processed in the English language.
7. The dual-mechanism model is a hybrid between the connectionist model approach of McClelland and the rule-based model of Chomsky.
8. There is some problems surfacing around the evidence initially used to support the dual-mechanism approach.
9. Language Processing relies more heavily on morphological decomposition in second language learners.
10. In native speakers non-productive derivational processing parallels that of irregular inflection

## Answers

The answers to the exercises are below the reference section.