A workbook is a type of textbook that has only practice questions and exercises. Workbooks are designed not to teach but to provide practice and to highlight areas which need more learning. A revision guide is a type of textbook that is used to remind the learner about the subject and give him/her extra practice, especially before an examination.
A textbook is usually lent to students by a school to accompany a course the school is teaching. Sometimes, especially at university, students have to buy the textbooks they need themselves or borrow them from a library.
Most textbooks are only published in printed format. However, some are now available online as electronic books.
Axiom
An axiom is a concept in logic. It is a statement which is assumed to be true without question, and which does not require proof. It is also known as a postulate (as in the parallel postulate). The axiom is to be used as the premise or starting point for further reasoning or arguments, usually in logic or in mathematics.
This means it cannot be proved within the discussion of a problem. So inside some discussion, it is thought to be true. There are many reasons why it has no proof. For example,
Logic can be used to find theorems from the axioms. Then those theorems can be used to make more theorems. This is often how math works. Axioms are important because logical arguments start with them.
Euclid of Alexandria was a Greek mathematician. Around the year 300 BC, he made the earliest list of axioms which we know of. The following are just a few of them:
Set
A set is an idea from mathematics. A set has "members" (also called "elements"). A set is defined by its members, so any two sets with the same members are the same (e.g., if set formula_1 and set formula_2 have the same members, then "formula_3").
A set cannot have the same member more than once. Membership is the only thing that matters. For example, there is no order or other difference among the members. "Anything" can be a member of a set, including sets themselves (though if a set is a member of itself, paradoxes such as Russell's paradox can happen).
Imagine the set is a bag.
Various things can be put into a bag. Later on, a good question would be if a certain thing is in the bag. Mathematicians call this "element of". Something is an element of a set, if that thing can be found in the respective bag. The symbol used for this is formula_4:
formula_5,
which means that formula_6 is in the bag formula_7 or formula_6 is an element of formula_7.
Unlike a bag, a set can contain at most one item of a given type. So for a set of fruits, it would make no difference if there is one orange, or if there are 10 oranges.
Like a bag, a set can also be "empty". The empty set is like an empty bag: it has nothing in it. The "empty set" is also called the "null set" and is represented by the symbol formula_10.
If we consider, say, some sets of American cars, e.g. a set of all Fords and a set of all Dodges, we may also wish to consider the whole set of American cars. In this case, the set of all American cars would be called a universe.
In other words, a universe is a collection of all the elements one wishes to consider in a given problem. The universe is usually named formula_11.
Two sets can be compared. This is like looking into two different bags. If they contain the same things, they are equal. No matter, in which order these things are.
"For example", if formula_12 and formula_13, the sets are the same.
When mathematicians talk about a set, they sometimes want to know how big a set is (or what is the cardinality of the set). They do this by counting how many elements are in the set (how many items are in the bag). For finite sets the cardinality is a simple number. The empty set has a cardinality of 0. The set formula_14 has a cardinality of 2.
Two sets have the same cardinality if we can pair up their elements—if we can join two elements, one from each set. The set formula_14 and the set formula_16 have the same cardinality. E.g, we could pair "apple" with "sun", and "orange" with "moon". The order does not matter. It is possible to pair all the elements, and none is left out. But the set formula_17 and the set formula_18 have different cardinality.
If we try to pair them up, we always leave out one animal.
At times cardinality is not a number. Sometimes a set has infinite cardinality. The set of all integers is a set with infinite cardinality. Some sets with infinite cardinality are bigger (have a bigger cardinality) than others. There are more real numbers than there are natural numbers, for example, which means we cannot pair up the set of integers and the set of real numbers, even if we worked forever.
If you can count the elements of a set, it is called a countable set. Countable sets include all sets with a finite number of members. Countable sets also include some infinite sets, such as the natural numbers. You can count the natural numbers with formula_19. The natural numbers are nicknamed "the counting numbers", since they are what we usually use to count things with.
An uncountable set is an infinite set that is impossible to count. If we try to count the elements, we will always skip some. It does not matter what step we take. The set of real numbers is an uncountable set. There are many other uncountable sets, even such a small interval like formula_20.[3]
If you look at the set formula_21 and the set formula_22, you can see that all elements in the first set are also in the second set.<br> We say: formula_23 is a subset of formula_24<br>
As a formula it looks like this:<br>
formula_25
In general, when all elements of set formula_26 are also elements of set formula_27, we call formula_26 a subset of formula_27:<br>
formula_30.<br>
It is usually read "formula_26 is contained in formula_27."
"Example:" Every Chevrolet is an American car. So the set of all Chevrolets is contained in the set of all American cars.
There are different ways to combine sets.
The intersection formula_33 of two sets formula_26 and formula_27 is a set that contains all the elements that are both in set formula_26 and in set formula_27 at the same time.<br>
"Example:" When formula_26 is the set of all cheap cars, and formula_27 is the set of all American cars, then formula_33 is the set of all cheap American cars.
The union formula_41 of two sets formula_26 and formula_27 is a set that contains all the elements that are in set formula_26 or in set formula_27.
This "or" is the "inclusive" disjunction, so the union also contains the elements, that are in set formula_26 "and" in set formula_27.
By the way, this means, that the intersection is a subset of the union:
formula_48.
"Example:" When formula_26 is the set of all cheap cars, and formula_27 is the set of all American cars, then formula_41 is the set of all cars, without all expensive cars that are not from America.
Complement can mean two different things:
formula_55 <br>
The universe formula_11 is the set of all things you speak about.<br>
"Example:" When formula_11 is the set of all cars, and formula_26 is the set of all cheap cars,<br> then formula_26 is the set of all expensive cars.
formula_64<br>
It is also called the relative complement of formula_26 in formula_27 .<br>
"Example:" When formula_26 is the set of all cheap cars, and formula_27 is the set of all American cars, then formula_64 is the set of all expensive American cars.
If you exchange the sets in the set difference, the result is different:<br>In the "example" with the cars, the difference formula_70 is the set of all cheap cars, that are not made in America.
Most mathematicians use uppercase "ITALIC" (usually Roman) letters to write about sets (such as formula_26, formula_27, formula_73). The things that are seen as elements of sets are usually written with lowercase Roman letters.
One way of showing a set is by a list of its members, separated by commas, included in braces.
"For example",