nyuuzyou/wikis huggingface dataset:
De Morgan's Laws (Set Theory)/Relative Complement/General Case

== Theorem ==
Let $S$ be a set.
Let $T$ be a subset of $S$.
Let $\powerset T$ be the power set of $T$.
Let $\mathbb T \subseteq \powerset T$. 
Then:
==== Complement of Intersection ====
==== Complement of Union ====
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Intersection

== Theorem ==
Let $S$ be a set.
Let $T$ be a subset of $S$.
Let $\powerset T$ be the power set of $T$.
Let $\mathbb T \subseteq \powerset T$. 
Then:
:$\ds \relcomp S {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \relcomp S H$
== Proof ==
== Sources ==
* : Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets
* : Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.6 \ \text{(c)}$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Union

== Theorem ==
Let $S$ be a set.
Let $T$ be a subset of $S$.
Let $\powerset T$ be the power set of $T$.
Let $\mathbb T \subseteq \powerset T$. 
Then:
:$\ds \relcomp S {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \relcomp S H$
== Proof ==
== Sources ==
* : Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets
* : Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.6 \ \text{(d)}$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Complement

== Theorem ==
Let $T_1, T_2$ be subsets of a universal set $\mathbb U$.
Let $\overline T_1$ denote the set complement of $T_1$.
Then:
=== Complement of Intersection ===
=== Complement of Union ===
=== General Case ===
=== Family of Sets ===
== Sources ==
* 
* 
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection

== Theorem ==
Let $T_1, T_2$ be subsets of a universal set $\mathbb U$.
Then:
:$\overline {T_1 \cap T_2} = \overline T_1 \cup \overline T_2$
where $\overline T_1$ is the set complement of $T_1$.
It is arguable that this notation may be easier to follow:
:$\map \complement {T_1 \cap T_2} = \map \complement {T_1} \cup \map \complement {T_2}$
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Demonstration by Venn Diagram ==
== Sources ==
* : Chapter $\text {II}$: Topological Spaces: $\S 8$. Notations and definitions of set theory
* : Introduction to Set Theory: $1$. Elementary Operations on Sets
* : Chapter $1$: Set Theory: $1.3$: Set operations
* : Chapter $1$: The Notation and Terminology of Set Theory: $\S 8 \beta$
* : $\S 1.6$: Set Identities and Other Set Relations
* : Appendix $\text{A}.1$: Sets and Subsets: Theorem $\text{A}.1 \ \text{(b)}$
* : Chapter $1$: Sets and mappings: $\S 1.2$: Sets: Exercise $6$
* 
* 
* : $\text {(x)}$
* 
* : $\S 2$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Corollary

== Theorem ==
Let $T_1, T_2$ be subsets of a universal set $\mathbb U$.
Then:
:$T_1 \cap T_2 = \overline {\overline T_1 \cup \overline T_2}$
== Proof ==
== Sources ==
* : $\text I$: Random Events: $1$. Relations among Random Events: Example $1.3$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Proof 2

== Theorem ==
== Proof ==
By definition of set equality:
:$\overline {T_1 \cap T_2} = \overline {T_1} \cup \overline {T_2}$
== Sources ==
* : Appendix $\text{A}.1$: Sets and Subsets: Problem Set $\text{A}.1$: $6$
* : $\S 2$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Venn Diagram

== Theorem ==
== Proof ==
Demonstration by Venn diagram:
:
$\overline T_1$ is depicted in yellow and $\overline T_2$ is depicted in red.
Their intersection, where they overlap, is depicted in orange.
Their union $\overline T_1 \cup \overline T_2$ is the total shaded area: yellow, red and orange.
As can be seen by inspection, this also equals the complement of the intersection of $T_1$ and $T_2$.
== Sources ==
* : Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.2$
* : Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson: Ponderable $1.2.1 \ \text{(c)}$
Category:De Morgan's Laws
Category:Examples of Venn Diagrams
De Morgan's Laws (Set Theory)/Set Complement/Complement of Union

== Theorem ==
Let $T_1, T_2$ be subsets of a universal set $\mathbb U$.
Then:
:$\overline {T_1 \cup T_2} = \overline T_1 \cap \overline T_2$
where $\overline T_1$ is the set complement of $T_1$.
It is arguable that this notation may be easier to follow:
:$\map \complement {T_1 \cup T_2} = \map \complement {T_1} \cap \map \complement {T_2}$
=== Corollary ===
== Proof 1 ==
== Proof 2 ==
== Demonstration by Venn Diagram ==
== Sources ==
* : Introduction to Set Theory: $1$. Elementary Operations on Sets
* : Chapter $1$: Set Theory: $1.3$: Set operations
* : Chapter $1$: The Notation and Terminology of Set Theory: $\S 8 \beta$
* : $\S 1.6$: Set Identities and Other Set Relations
* : Appendix $\text{A}.1$: Sets and Subsets: Theorem $\text{A}.1 \ \text{(a)}$
* : Chapter $1$: Sets and mappings: $\S 1.2$: Sets: Exercise $6$
* 
* 
* : $\text {(x)}$
* 
* : $\S 2$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Corollary

== Theorem ==
Let $T_1, T_2$ be subsets of a universal set $\mathbb U$.
Then:
:$T_1 \cup T_2 = \overline {\overline T_1 \cap \overline T_2}$
== Proof ==
== Sources ==
* : $\text I$: Random Events: $1$. Relations among Random Events: Example $1.3$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Proof 2

== Theorem ==
== Proof ==
By definition of set equality:
:$\overline {T_1 \cup T_2} = \overline {T_1} \cap \overline {T_2}$
== Sources ==
* : Chapter $\text {II}$: Topological Spaces: $\S 8$. Notations and definitions of set theory
* : Chapter $1$: Theory of Sets: $\S 3$: Set Operations: Union, Intersection and Complement: Theorem $3.1$
* : Appendix $\text{A}.1$: Sets and Subsets: Theorem $\text{A}.1 \ \text{(a)}$
* : $\S 2$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Venn Diagram

== Theorem ==
== Proof ==
Demonstration by Venn diagram:
:
$\overline T_1$ is depicted in yellow and $\overline T_2$ is depicted in red.
Their intersection, $\overline T_1 \cap \overline T_2$, is depicted in orange.
As can be seen by inspection, this also equals the complement of the union of $T_1$ and $T_2$.
== Sources ==
* : Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson: Ponderable $1.2.1 \ \text{(d)}$
Category:De Morgan's Laws
Category:Examples of Venn Diagrams
De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection

== Theorem ==
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets, all of which are subsets of a universal set $\mathbb U$.
Then:
:$\ds \map \complement {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \map \complement {S_i}$
== Proof ==
== Also see ==
* De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union
== Sources ==
* : $\S 1.1$: Rings and Algebras
* : Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets: Theorem $4.2$
Category:De Morgan's Laws
Category:Indexed Families
De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union

== Theorem ==
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets, all of which are subsets of a universal set $\mathbb U$.
Then:
:$\ds \map \complement {\bigcup_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \map \complement {S_i}$
== Proof ==
== Sources ==
* : $\S 1.1$: Rings and Algebras: $(1.1.1)$
* : Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets: Theorem $4.1$
Category:De Morgan's Laws
Category:Indexed Families
De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Intersection

== Theorem ==
Let $\mathbb T$ be a set of sets, all of which are subsets of a universal set $\mathbb U$.
Then:
:$\ds \map \complement {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \map \complement H$
== Proof ==
== Sources ==
* : Introduction to Set Theory: $1$. Elementary Operations on Sets
* : $\S 1.8$: Collections of Sets: Exercise $5$
* : $\S 7.1 \ \text{(i)}$: Unions and Intersections
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Union

== Theorem ==
Let $\mathbb T$ be a set of sets, all of which are subsets of a universal set $\mathbb U$.
Then:
:$\ds \map \complement {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \map \complement H$
== Proof ==
== Sources ==
* : Introduction to Set Theory: $1$. Elementary Operations on Sets
* : $\S 1.8$: Collections of Sets: Exercise $5$
* : $\S 7.1 \ \text{(ii)}$: Unions and Intersections
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Difference

== Theorem ==
Let $S, T_1, T_2$ be sets.
Let:
:$T_1 \cap T_2$ denote set intersection
:$T_1 \cup T_2$ denote set union.
Then:
=== Difference with Intersection ===
=== Difference with Union ===
=== General Case ===
=== Family of Sets ===
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection

== Theorem ==
Let $S, T_1, T_2$ be sets.
Then:
:$S \setminus \paren {T_1 \cap T_2} = \paren {S \setminus T_1} \cup \paren {S \setminus T_2}$
where:
:$T_1 \cap T_2$ denotes set intersection
:$T_1 \cup T_2$ denotes set union.
=== Illustration by Venn Diagram ===
=== Corollary ===
== Proof ==
By definition of set equality:
: $S \setminus \paren {T_1 \cap T_2} = \paren {S \setminus T_1} \cup \paren {S \setminus T_2}$
== Sources ==
* : Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $2 \ \text{(e)}$
* : Exercise $1.1: \ 8 \ \text{(h)}$
* : Introduction to Set Theory: $1$. Elementary Operations on Sets
* : Chapter $1$. Sets: Exercise $6$
* : Chapter $1$: The Notation and Terminology of Set Theory: $\S 8 \ \text{(e)}$
* : $\S 1.6$: Set Identities and Other Set Relations: Theorem $6.1$
* : $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
* : $\S 2$: Problem $1 \ \text{(iv)}$
* : $\S 1.1$: Sets and Functions
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection/Corollary

== Corollary to De Morgan's Laws: Difference with Intersection ==
Let $S, T_1, T_2$ be sets.
Suppose that $T_1 \subseteq S$.
Then:
:$S \setminus \paren {T_1 \cap T_2} = \paren {S \setminus T_1} \cup \paren {T_1 \setminus T_2}$
== Proof ==
Category:Set Difference
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection/Venn Diagram

== Theorem ==
== Proof ==
Demonstration by Venn diagram:
:
The area in blue and magenta is the set difference of $S$ with $T_1$
The area in orange and magenta is set difference of $S$ with $T_2$
The complete shaded area is the set difference of $S$ with the intersection of $T_1$ and $T_2$.
It is also seen to be the union of the set difference of $S$ with $T_1$ and the set difference of $S$ with $T_2$.
== Sources ==
* : Exercise $1.1: \ 8 \ \text{(h)}$
Category:De Morgan's Laws
Category:Examples of Venn Diagrams
De Morgan's Laws (Set Theory)/Set Difference/Difference with Union

== Theorem ==
Let $S, T_1, T_2$ be sets.
Then:
:$S \setminus \paren {T_1 \cup T_2} = \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$
where:
:$T_1 \cap T_2$ denotes set intersection
:$T_1 \cup T_2$ denotes set union.
=== Illustration by Venn Diagram ===
== Proof ==
By definition of set equality:
:$S \setminus \paren {T_1 \cup T_2} = \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$
== Sources ==
* : Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $2 \ \text{(e)}$
* : $\S 1.1$: Theorem $1.8$
* : Introduction to Set Theory: $1$. Elementary Operations on Sets
* : $\S 1.6$. Difference and complement: Example $21$
* : Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.4 \ \text{(a)}$
* : Chapter $1$: The Notation and Terminology of Set Theory: $\S 8 \ \text{(f)}$
* : $\S 1.6$: Set Identities and Other Set Relations: Exercise $2 \ \text{(h)}$
* : $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
* : $\S 2$: Problem $1 \ \text{(v)}$
* : $\S 1.1$: Sets and Functions
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Difference/Difference with Union/Venn Diagram

== Theorem ==
== Proof ==
Demonstration by Venn diagram:
:
The area in orange and red is the set difference of $S$ with $T_1$
The area in orange and yellow is set difference of $S$ with $T_2$
The area in orange only is the set difference of $S$ with the union of $T_1$ and $T_2$.
It is also seen to be the intersection of the set difference of $S$ with $T_1$ and the set difference of $S$ with $T_2$.
== Sources ==
* : $\S 1.1$: Theorem $1.8$
* : $\S 1.6$. Difference and complement: Example $21$
* : Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.4 \ \text{(a)}$
Category:De Morgan's Laws
Category:Examples of Venn Diagrams
De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Intersection

== Theorem ==
Let $S$ and $T$ be sets.
Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$.
Then:
:$\ds S \setminus \bigcap_{i \mathop \in I} T_i = \bigcup_{i \mathop \in I} \paren {S \setminus T_i}$
where:
:$\ds \bigcup_{i \mathop \in I} T_i := \set {x: \exists i \in I: x \in T_i}$
that is, the union of $\family {T_i}_{i \mathop \in I}$.
== Proof ==
Suppose:
:$\ds x \in S \setminus \bigcap_{i \mathop \in I} T_i$
Note that by Set Difference is Subset we have that $x \in S$ (we need this later).
Then:
Therefore:
:$\ds S \setminus \bigcap_{i \mathop \in I} T_i = \bigcup_{i \mathop \in I} \paren {S \setminus T_i}$
== Sources ==
* : Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $3 \ \text{(b)}$
* : $1$ Set Theory: $1$. Sets and Functions: $1.2$: Operations on sets: $(4)$
* : Notation and Terminology
* : Appendix $\text{A}.2$: Boolean Operations: Problem $\text{A}.2.2$
Category:De Morgan's Laws
Category:Indexed Families
De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Union

== Theorem ==
Let $S$ and $T$ be sets.
Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$.
Then:
:$\ds S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \mathop \in I} \paren {S \setminus T_i}$
where:
:$\ds \bigcap_{i \mathop \in I} T_i := \set {x: \forall i \in I: x \in T_i}$
that is, the intersection of $\family {T_i}_{i \mathop \in I}$.
== Proof ==
Suppose:
:$\ds x \in S \setminus \bigcup_{i \mathop \in I} T_i$
Note that by Set Difference is Subset we have that $x \in S$ (we need this later).
Then:
Therefore:
:$\ds S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \mathop \in I} \paren {S \setminus T_i}$
== Sources ==
* : Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $3 \ \text{(b)}$
* : $1$ Set Theory: $1$. Sets and Functions: $1.2$: Operations on sets: $(3)$
* : Notation and Terminology
* : Appendix $\text{A}.2$: Boolean Operations: Problem $\text{A}.2.2$
Category:De Morgan's Laws
Category:Indexed Families
De Morgan's Laws (Set Theory)/Set Difference/General Case

== Theorem ==
Let $S$ and $T$ be sets.
Let $\powerset T$ be the power set of $T$.
Let $\mathbb T \subseteq \powerset T$.
Then:
==== Difference with Intersection ====
==== Difference with Union ====
== Sources ==
* : $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts: Exercise $9$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection

== Theorem ==
Let $S$ and $T$ be sets.
Let $\powerset T$ be the power set of $T$.
Let $\mathbb T \subseteq \powerset T$.
Then:
:$\ds S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$
where:
:$\ds \bigcap \mathbb T := \set {x: \forall T' \in \mathbb T: x \in T'}$
that is, the intersection of $\mathbb T$
== Proof ==
== Caution ==
It is tempting to set up an argument to prove the general case using induction. While this works, and is a perfectly valid demonstration for an elementary student in how such proofs are crafted, such a proof is inadequate as it is valid only when $\mathbb T$ is finite.
The proof as given above relies only upon De Morgan's laws as applied to predicate logic. Thus the uncountable case has been reduced to a result in logic. 
However, for better or worse, the following is an example of how one might achieve this result using induction.
== Proof by Induction ==
== Sources ==
* : Chapter $\text{I}$: Sets and Functions: Unions and Intersections: Theorem $2 \ \text{(iv)}$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection/Proof

== Theorem ==
== Proof ==
Suppose:
:$\ds x \in S \setminus \bigcap \mathbb T$
Note that by Set Difference is Subset we have that $x \in S$ (we need this later).
Then:
Therefore:
:$\ds S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$
== Sources ==
* : Chapter $\text I$: Sets and Functions: Exercise $\text{D}$
Category:De Morgan's Laws
De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union

== Theorem ==
Let $S$ and $T$ be sets.
Let $\powerset T$ be the power set of $T$.
Let $\mathbb T \subseteq \powerset T$.
Then:
:$\ds S \setminus \bigcup \mathbb T = \bigcap_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$
where:
:$\ds \bigcup \mathbb T := \set {x: \exists T' \in \mathbb T: x \in T'}$
that is, the union of $\mathbb T$.
== Proof ==
Suppose:
:$\ds x \in S \setminus \bigcup \mathbb T$
Note that by Set Difference is Subset we have that $x \in S$ (we need this later).
Then:
Therefore:
:$\ds S \setminus \bigcup \mathbb T = \bigcap_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$
== Caution ==
It is tempting to set up an argument to prove the general case using induction. While this works, and is a perfectly valid demonstration for an elementary student in how such proofs are crafted, such a proof is inadequate as it is valid only when $\mathbb T$ is finite.
The proof as given above relies only upon De Morgan's laws as applied to predicate logic. Thus the uncountable case has been reduced to a result in logic. 
However, for better or worse, the following is an example of how one might achieve this result using induction.
== Proof by Induction ==
== Sources ==
* : Chapter $\text{I}$: Sets and Functions: Unions and Intersections: Theorem $2 \ \text{(iii)}$
Category:De Morgan's Laws
De Méré's Paradox

== Paradox ==
Which of these two is more probable?
* Getting at least one six with four throws of a die;
* Getting at least one double six with 24 throws of a pair of dice?
The self-styled believed the two to be equiprobable, based on the following reasoning:
# A pair of sixes on a single roll of two dice is the same probability as that of rolling two sixes on two rolls of one die.
# The probability of rolling two sixes on two rolls is $1/6$ as likely as one six in one roll.
# To make up for this, a pair of dice should therefore be rolled six times for every one roll of a single die in order to get the same chance of a pair of sixes.
# Therefore, rolling a pair of dice six times as often as rolling one die should equal the probabilities.
# So rolling 2 dice 24 times should result in as many double sixes as getting 1 six throwing 4 dice.
However, betting on getting 2 sixes when rolling 24 times, he lost consistently.
== Resolution ==
As throwing a die is an experiment with a finite number of equiprobable outcomes, we can use the Classical Probability Model.
There are six sides to a die, so there is $1/6$ probability for a six to turn up in one throw.
That is, by Elementary Properties of Probability Measure there is a $1 - \frac 1 6 = \frac 5 6$ probability for a six not to turn up.
When you throw a die $4$ times, by Probability of Independent Events Not Happening, there is $\left({1 - \frac 1 6}\right)^4 = \left({\frac 5 6}\right)^4$ probability of a six not turning up at all.
So by Probability of Occurrence of At Least One Independent Event, there is a probability of $1 - \left({\frac 5 6}\right)^4$ of getting at least one six with $4$ rolls of a die.
Doing the arithmetic gives you a probability of $> 0.5$, or in favour of a six appearing in 4 rolls.
Now when you throw a pair of dice, from the definition of independent events, there is a $\left({\frac 1 6}\right)^2 = \frac 1 {36}$ probability of a pair of sixes appearing.
That is, by Elementary Properties of Probability Measure, $\frac {35} {36}$ for a pair of sixes not appearing.
So by Probability of Occurrence of At Least One Independent Event there is a probability of $1 - \left({\frac {35} {36}}\right)^{24}$ of getting at least one pair of sixes with $24$ rolls of a pair of dice.
Doing the arithmetic gives you a probability of $< 0.5$, or in favour of a pair of sixes not appearing in 24 rolls.
== Consequence ==
This is a veridical paradox.
Counter-intuitively, the odds are distributed differently from how they would be expected to be.
== Historical Note ==
== Sources ==
* : $1$: Events and probabilities: $1.5$: Discrete sample spaces: Exercise $16$
* : $\S 6.1$
* 
Category:Veridical Paradoxes
Category:Probability Theory
De Méré's Paradox/Historical Note

== Historical Note on De Méré's Paradox ==
The first raised this question in the $17$th century.
He believed the two events described should have the same probabilities.
Empirical investigation (in other words: he found he was losing more money than he believed he ought to have been doing) caused him to rethink this.
Hence he posed this problem to his friend mathematician , who solved it.
== Sources ==
* : $1$: Events and probabilities: $1.5$: Discrete sample spaces: Exercise $16$
Category:Historical Notes
De Méré's Problem of Points

== Classic Problem ==
Two gamblers are playing a game of dice.
The object is to win as many points as possible to win a game.
If they abandon the game before it has finished, how should the stakes be divided between them?
== Solution ==
== Historical Note ==
== Sources ==
* : Chapter $\text{V}$: "Greatness and Misery of Man"
Category:Probability Theory
Category:Classic Problems
De Polignac's False Conjecture

== Famous False Conjecture ==
Every odd number greater than $1$ can be expressed as the sum of a power of $2$ and a prime.
=== Investigation ===
== Refutation ==
The smallest integer for which this conjecture fails is $127$:
== Also see ==
* Definition:Obstinate Number
== Historical Note ==
== Sources ==
* : $127$
* : $127$
Category:Famous False Conjectures
Category:Obstinate Numbers
Category:De Polignac's False Conjecture
De Polignac's False Conjecture/Historical Note

== Historical Note on De Polignac's False Conjecture ==
 proposed his false conjecture in $1848$, claiming that he had verified it up to $3$ million.
This was clearly an exaggeration, as the conjecture fails for the modestly small number $127$.
, in his of $1986$, mentions that this topic is discussed by in , no. $6$.
This needs to be corroborated.
== Sources ==
* : $127$
* : $127$
Category:Historical Notes