SM291 Definitions

At most countable

A set A is at most countable if it is finite or countable

Countably infinite

A set A is countably infinite or simply countable, if A ~ N. An inifinite set that is not countable is called uncountable

Finite

A set A is finite, if it is the empty set or there exists n in N such that A ~ {1,2,...,n}.

Injective

Let A and B be sets and let ƒ: A→B be a function. The function ƒ is injective (also called one-to-one or monic) if x≠y impiles ƒ(x)≠ƒ(y) for all x,y ∈A; equivalently if ƒ(x)=ƒ(y) implies x=y for all x,y ∈A.

Surjective

Let A and B be sets and let ƒ: A→B be a function. The function ƒ is surjective (also called onto or epic) if for every b∈B, there exists some a∈A such that ƒ(a) ∈ A; equivalently if ƒ(A) = B.

Bijective

Let A and B be sets and let ƒ: A→B be a function. ƒ is bijective if it is both injective and surjective.

Right Inverse

Let A and B be sets, and let ƒ: A→B and g: B→A be functions. The function g is a _____ for ƒ if (f o g) = 1_B

Left Inverse

Let A and B be sets, and let ƒ: A→B and g: B→A be functions. The function g is a _____ for ƒ if (g o f) = 1_A

Inverse

Let A and B be sets, and let ƒ: A→B and g: B→A be functions. The function g is an inverse for ƒ if it is both a right and left inverse. If ƒ has an inverse, g can also be denoted ƒ^(-1): B →A

Image of a set under a function

Let A and B be sets, and let ƒ: A→B be a function. Let P ⊆ A. The ____ of P under ƒ, denoted ƒ(P), is the set defined by ƒ(P) = {b∈B: b = ƒ(p) for some p∈P}. The range of ƒ (also called the _____ of ƒ) is the set ƒ(A)

Inverse image of a set under a function

Let A and B be sets, and let ƒ: A→B be a function. Let Q ⊆ B. The ____ of Q under ƒ, denoted ƒ^(-1)(Q), is the set defined by ƒ^(-1)(Q) = {a∈A: ƒ(a) ∈ Q}. The range of ƒ (also called the _____ of ƒ) is the set ƒ(A)

Functions (Maps)

Let A and B be sets. A _____ (also called a _____) ƒ from A to B, denoted ƒ:A→B is a subset F ⊆ AxB such that for each a∈A, there is one and only one pair in F of the form (a,b). The set A is called the domain of ƒ and the set B is called the codomain of ƒ.

Relations

Let A and B be sets. A relation R from A to B is a subset R ⊆AxB. If a∈A and b∈B, we write aRb if (a,b)∈R, and a(strikethruR)b if (a,b)∉R. A relation "on" A is a relation from A to A.

Cartesian product of two sets

Let A and B be sets. The _____ (also called the _____ _____) of A and B, denoted AxB is the set AxB={(a,b) : a∈A and b∈B}, where (a,b) denotes an ordered pair.

Difference of sets

Let A and B be sets. The _____ (also called the set _____) of A and B, denoted A\B, is the set defined by A\B = {x: x∈A and x∉B}

Intersection of sets

Let A and B be sets. The _____ of A and B, denoted A∩B, is the set defined by A∩B = {x: x∈A and x∈B}

Union of sets

Let A and B be sets. The _____ of A and B, denoted A∪B, is the set defined by A∪B = {x : x∈A or x∈B}

Subsets

Let A and B be sets. The set A is a ______ of the set B, denoted A⊆B, if x∈A implies x∈B. If A is not a subset of B, we write A⊄B

Reflexive

Let A be a non-empty set, and R be a relation on A. The relation R is reflexive if xRx for all x∈A

Symmetric

Let A be a non-empty set, and R be a relation on A. The relation R is symmetric if xRy implies yRx for all x,y∈A

Transitive

Let A be a non-empty set, and R be a relation on A. The relation R is transitive if xRy and yRz imply xRz, for all x,y,z∈A.

Anti-symmetric

Let A be a non-empty set, and R be a relation on A. if aRb and bRa implies a=b, we call this anti-symmetric

Equivalence class

Let A be a non-empty set, and let ~ be an equivalence relation on A. The relation classes of A with respect to ~ are called equivalence classes. A relation class of x with respect to R, denoted R[x], is the set defined by R[x] = {y∈A : xRy}.

Quotient space

Let A be a set and R be a relation The quotient space of a relation. The quotient space of A/R, denoted as such, is the set of all equivalence classes in A.

Equivalence relation

Let A be a set and let ~ be a relation on A. The relation ~ is an equivalence relation if it is reflexive, symmetric, and transitive

Power Set

Let A be a set. The _____ of A, denoted P(A) is the set defined by P(A) = {X : X ⊆ A}

Composition of functions

Let A, B, C be sets, and let ƒ: A →B and g: B → C be functions. The _____ of f and g is the function g o ƒ: A →C defined by (g o ƒ)(x) = g(f(x))

Intersection of Family of Sets

Let F be a family of sets. The _____ of the sets in F, denoted ∩_(X∈F) X, is defined as follows. If F≠{} , then ∩_(X∈F) X = {x : x∈ A for all A ∈F}. If F = {} then ∩_(X∈F) X is not defined. If F = {A_i}_(i∈I) is indexed by a set I, then we write ∩_(X∈F) X = {x : x∈ A_i for all i ∈ I}

Union of Family of Sets

Let F be a family of sets. The _____ of the sets in F, denoted ∪_(X∈F) X, is defined as follows. If F≠{} , then ∪_(X∈F) X = {x : x∈ A for some A ∈F}. If F = {} then ∪_(X∈F) X = {}. If F = {A_i}_(i∈I) is indexed by a set I, then we write ∪_(X∈F) X = {x : x∈ A_i for some i ∈ I}

Family of Sets

Let F be a set. The set F is called a _____ if all the elements of F are sets. The family of sets F is indexed by I, denoted F = {A_i}_(i∈I) if there is a non-empty set I such that there is an element A_i ∈ F for each i∈I, and that every element of F equals A_i for exactly one i∈I **Note: A_i is A subscript i, which does not come across in quizlet**

rational number

Let x be a real number. The number x is a rational number if there exist integers n and m such that m ≠0 and x = n/m. If x is not a rational number then it is an irrational number.

{2,3,4,...} ~ N

Proof: f: N -> {2,3,4,..} defined by f(n) = n+1. This is bijection, thus {2,3,4,...}~ N

If g o f is injective, then f is injective

See homework for proof

If g o f is surjective, then g is surjective

See homework for proof

if gof is bijective, then f is injective and g is surjective

See homework for proof

For any real numbers a<b we have (a,b) ~(0,1) (where (a,b) = {x in R : a<x<b}).

See notebook for proof

Let B be the set of all functions f: N → {0,1} (so, B is the set of all "binary sequences"). Then B is uncountable.

See notebook for proof

NxN is countable

See notebook for proof

Q is countable

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THM: Let A be a subset of N. Then A is finite or countable

See notebook for proof

THM: Suppose that X is an infinite set. Then X has a countable subset

See notebook for proof

Z is countable

See notebook for proof

Enumeration

Suppose A is a countable set, and let f: N → A be a bijection. Denoting f(n) by a_n, we have that A = {a_1,a_2,...} = {a_n : n in N}. We say this is an enumeration of A

Identity map

The _____ on A is the function 1_A : A→A defined by 1_A(x) = x for all x∈A

a divides b (a|b)

There exists some integer k such that b = k*a

Cardinality

We say that two sets A and B are eqivalent, or that they have the same cardinality, if there exists a bijection f: A→B. We then write A∼B.

a ≡ b mod c

c | a-b

THM: Let A be an infinite set. The following statements are equivalent.

i) A is countable ii) There exists a surjection f: N →A iii) there exists an injection g: A →N corollary: If A is countable and f: A→B is onto, then B is at most countable

THM: Suppose A,B,C are sets. Let ~ mean two sets have the same cardinality

i) A∼A ii) A∼B => B∼A iii) A∼B and B∼C => A∼C