lacunary - Mathnotes

Set Theory

Definition: Set (also: collection, family) \@{set}

A set is a collection of objects, considered as a whole.

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Definition: Element (also: member, point) \@{element}

The objects that make up a set are called its elements or its members.

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Definition: Membership criterion \@{membership-criterion}

The membership criterion for a set X is a statement of the form x∈X⟺P(x), where P(x) is a proposition that is true for precisely for those objects x that are elements of X, and no others.

Axiom: Axiom of Extensionality \@{axiom-of-extensionality}

Two sets are equal if and only if they have the same elements. Formally, for any sets A and B:

A=B⟺(∀x)(x∈A⟺x∈B)

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Definition: subset \@{subset}

If X and Y are sets such that every element of X is also an element of Y, then we say X is a subset of Y, denoted as X⊂Y. Formally,

X⊂Y⟺(∀x)(x∈X⟹x∈Y)

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Definition: superset \@{superset}

If X and Y are sets such that every element of Y is also an element of X, then we say X is a superset of Y, denoted as X⊃Y. This is the same as Y⊂X.

Definition: Empty Set \@{empty-set}

The empty set is the set that has no elements and is denoted {} or ∅. It is the unique set with @cardinality of 0.

Theorem \@{set-equality-via-subset-inclusion}

Two sets X and Y are equal if and only if X is a subset of Y and Y is a subset of X.

Proof \@{proof-of-set-equality-via-subset-inclusion}

Suppose X and Y are sets with X⊂Y and Y⊂X. Now, suppose x∈X. Then, x∈Y. Conversely, suppose y∈Y. Then y∈X. Thus, (∀x)(x∈X⟺x∈Y), and X=Y.

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Definition: Function (also: mapping, map) \@{function}

Consider two sets, A and B, whose elements may be any objects whatsoever, and suppose that with each element x of A there is associated, in some manner, any element of B, which we denote by f(x). Then f is said to be a function from A to B.

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Definition: Domain of Definition \@{domain-of-definition}

If f is a function from the set A to the set B, the set A is called the domain of f.

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Definition: Value \@{value}

If f is a function from the set A to the set B, the elements f(x)∈B are called the values of f.

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Definition: Injective (also: one-to-one, injection) (also defines: Surjective, Bijective) \@{injective}

A function f:A→B is

Injective (also: one-to-one, injection) if for all a,a′∈A,a≠a′⟹f(a)≠f(a′).
Surjective (also: onto, surjection) if for every b∈B there is an a∈A with f(a)=b.
Bijective (also: bijection) if f is both injective and surjective.
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Definition: Sequence \@{sequence}

A sequence is a function f defined on the set J of all positive integers.

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Note \@{sequence-notation}

If f(n)=xn, for n∈J, it is customary to denote the sequence f by the symbol {xn}, or sometimes by x1,x2,x3,….

Definition: Term \@{term}

If f is a sequence denoted as {xn}, the values of f, that is, the elements xn, are called the terms of the sequence.

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Note \@{sequence-terms-not-distinct}

The terms of a sequence need not be distinct.

Definition: Inverse Image (also: pre-image, preimage) \@{inverse-image}

If f:A→B and E⊂(B), then f−1(E) denotes the set of all x∈A such that f(x)∈E. We call f−1(E) the inverse image of E under f.

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De Morgan's Laws

Theorem \@{complement-of-union-is-intersection-of-complements}

The complement of a union is equal to the intersection of complements.

Proof \@{proof-of-complement-of-union-is-intersection-of-complements}

Let A and B be sets. We want to show that

(A∪B)c=Ac∩Bc.

Suppose x∈(A∪B)c. Then, if x∈A or x∈B, then x∈A∪B and x∉(A∪B)c, a contradiction. Therefore, x∉A and x∉B. That is, x∈Ac and x∈Bc, therefore x∈Ac∩Bc.

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Theorem \@{complement-of-intersection-is-union-of-complements}

The complement of an intersection is equal to the union of complements.

Proof \@{proof-of-complement-of-intersection-is-union-of-complements}

Let A and B be sets. We want to show that

(A∩B)c=Ac∪Bc.

Suppose x∈(A∩B)c. Then, x is not in A∩B, that is, x is either not in A or it is not in B or it is in neither. If x is not in A, then it is in Ac, and therefore it is in Ac∪Bc. The same approach works with B, and therefore x∈Ac∪Bc, and we have shown (A∩B)c=Ac∪Bc.

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