Set theory

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Basic Definition

The term "set" can be thought as a well-defined collection of objects. In set theory, These objects are often called "elements".

• We usually use capital letters for the sets, and lowercase letters for the elements.
• If an element $a$ belongs to a set $A$, we can say that "$a$ is a member of the set $A$", or that "$a$ is in $A$", or simply write $a\in A$.
• Similarly, if $a$ is not in $A$, we would write $a\notin A$.

(Example) $x\in \mathbb {R}$. In this case, $x$ is the element and $\mathbb {R}$ is the set of all real numbers.

Notation

Example of a common notation style for the definition of a set:

• $S$ is a set
• $P(x)$ is a property (The elements of $S$ may or may not satisfy this property)
• Set $A$ can be defined by writing:

$A=\{x\in S\mid P(x)\}$

This would read as "the set of all $x$ in $S$, such that $P$ of $x$."

Elements

There are two ways that we could show which elements are members of a set: by listing all the elements, or by specifying a rule which leaves no room for misinterpretation. In both ways we will use curly braces to enclose the elements of a set. Say we have a set $A$ that contains all the positive integers that are smaller than ten. In this case we would write $A=\{1,2,3,4,5,6,7,8,9\}$. We could also use a rule to show the elements of this set, as in $A=\{a:{\text{ }}a{\text{ positive integer less than 10}}\}$.

In a set, the order of the elements is irrelevant, as is the possibility of duplicate elements. For example, we write $X=\{1,2,3\}$ to denote a set $X$ containing the numbers 1, 2 and 3. Or,$X=\{1,2,3\}=\{3,2,1\}=\{1,1,2,3,3\}$.

Subsets

A is a subset of B. $\left(A\subseteq B\right)$

Formal universal conditional statement: "set A is a subset of a set B"

• $A\subseteq B\Leftrightarrow \forall x$, if $x\in A$, then $x\in B.$

Negation:

• $A\nsubseteq B\Leftrightarrow \exists x$ such that $x\in A$ and $x\notin B.$

If and only if: 
    for all $x$:
        If: ($x$ is an element of A)
        then: ($x$ is an element of B)
then: 
    set A is a subset of set B

Truth Table Example:

$x\in A$	$x\in B.$	if $x\in A$, then $x\in B.$	$x\in A$ and
1	1	1	0
1	0	0	1
0	1	1	0
0	0	1	0

A proper subset of a set is a subset that is not equal to its containing set. Thus

A is a proper subset of B $\iff$

Set Identities

Let all sets referred to below be subsets of a universal set U.

(a) A ∪ ∅ = A and (b) A ∩ U = A.

5. Complement Laws:

(a) A ∪ A c = U and (b) A ∩ A c = ∅.

6. Double Complement Law:

(A c ) c = A.

7. Idempotent Laws:

(a) A ∪ A = A and (b) A ∩ A = A.

8. Universal Bound Laws:

(a) A ∪ U = U and

(b) A ∩ ∅ = ∅.

$A\cap U=A$	$A\cap U=A$

Identity	For all sets A
Identity Laws:	$A\cup \emptyset =A$	$A\cap U=A$ $A\cap U=A$
Complement Laws:	$A\cup A^{c}=U$ $A\cap A^{c}=\emptyset$
Double Complement Law:	$\left(A^{c}\right)^{c}=A$
Idempotent Laws:	$A\cup A=A$ $A\cap A=A$
Universal Bound Laws:

Identity	For all sets A and B
Commutative Laws:	$A\cup B=B\cup A$$A\cap B=B\cap A$

2. Associative Laws: For all sets A, B, and C,

(a) (A ∪ B) ∪ C = A ∪ (B ∪ C) and

(b) (A ∩ B) ∩ C = A ∩ (B ∩ C).

3. Distributive Laws: For all sets, A, B, and C,

(a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and

(b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

For all sets A and B,

De Morgan’s Laws:

(a) (A ∪ B) c = A c ∩ B c and (b) (A ∩ B) c = A c ∪ B c .

Absorption Laws:

(a) A ∪ (A ∩ B) = A and (b) A ∩ (A ∪ B) = A.

Set Difference Law:

• A − B = A ∩ B c .

Complements of U and ∅:

• $U^{c}=\emptyset$
• $\emptyset ^{c}=U$

Cardinality

The cardinality of a set is the number of elements in the set. The cardinality of a set $A$ is denoted $|A|$

Types of Sets by Cardinality

A set can be classified as finite, countable, or uncountable.

• Finite Sets are sets that have finitely many elements, $A=\{1,2,3\}$ is a finite set of cardinality 3.
• Countable Sets are sets that have as many elements as the set of natural numbers, $|\mathbb {Q} |=|\mathbb {N} |$ so the set of rational numbers is countable.
• Uncountable Sets are sets that have more elements than the set of natural numbers, $|\mathbb {R} |>|\mathbb {N} |$ so the set of real numbers is uncountable.

Common Sets of Numbers

Diagram to demonstrate the number systems ℝ, ℚ, ℤ and ℕ as sub-sets of each other.

• $\mathbb {N}$ is the set of Naturals
• $\mathbb {Z}$ is the set of Integers
• $\mathbb {Q}$ is the set of Rationals
• $\mathbb {C}$ is the set of Complex Numbers

Partitions of Sets

In many applications of set theory, sets are divided up into non-overlapping (or disjoint) pieces. Such a division is called a partition.

Two sets are called disjoint if, and only if, they have no elements in common.

A and B are disjoint ⇔ A ∩ B = ∅.

Sets $A_{1},A_{2},A_{3},\ldots$ are mutually disjoint (or pairwise disjoint or nonoverlapping)

if, and only if, no two sets $A_{i}$ and $A_{j}$ with distinct subscripts have any elements in

common. More precisely, for all $i,j=1,2,3,\ldots$

$A_{i}\cap A_{j}=\emptyset$ whenever $i\not =j$.

Power Sets

The power set of a set A is all possible subsets of A, including A itself and the empty set. Which can be represented:

$$P_{(A)}=\{\emptyset ,\{1\},\{2\},\ldots \}$$

For the set $A=\{1,2,3\}$

$$P_{(A)}=\{\emptyset ,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$$

Learning resources

Wikiversity

• Introduction_to_set_theory/Lecture_1

Wikipedia

• Set theory category