Math 117 Exam 2 questions with 100% correct answers

Sequence

Set of elements written in a row

Ex: a₁, a₂, a₃, ..., an

Infinite Sequence

Am, Am+1, Am+2, ...

Summation Notation

Product Notation

Factorial Notation

Properties of Summations

Well Ordering Principle

All sets with one or more integers all of which are greater than some fixed integer have a least element

Subsets

A⊆B ↔ ∀x, if x∈A then x∈B

Set Equality

A=B ↔ A⊆B and B⊆A

Difference

B-A ↔ x∈B and x∉A

Null Set

∅

Disjoint

A∩B = ∅

Power Set

P(A) - the set of all subsets of A

note: ∅ is a subset of every set equal to 0.

Cartesian Products

Inclusion of Intersection

A∩B ⊆ A and A∩B ⊆ B

Inclusion in Union

A ⊆ A∪B and B ⊆ A∪B

Transitive Property of Subsets

A ⊆ B and B ⊆ C, then A ⊆ C

Commutative Laws

(a) A∪B = B∪A

(b) A∩B = B∩A

Associative Laws

(a) (A∪B)∪C = A∪(B∪C)

(b) (A∩B)∩C = A∩(B∩C)

Distributive Laws

(a) A∪(B∪C) = (A∪B)∩(A∪C)

(b) A∩(B∪C) = (A∩B)∪(A∩C)

Identity Laws

(a) A∪∅ = A

(b) A∩U = A

Complement Laws

(a) A∪A° = U

(b) A∩A° = ∅

Double Complement Law

(a) (A°)° = A

Indempotent Laws

(a) A∪A = A

(b) A∩A = A

Universal Bound Laws

(a) A∪U = U

(b) A∩∅ = ∅

De Morgan's Law

(a) (A∪B)° = A°∩B°

(b) (A∩B)° = A°∪B°

Absorption Laws

(a) A∪(A∩B) = A

(b) A∩(A∪B) = A

Relation

xRy ↔ (x,y)∈R

Function

Property 1: For every element x in A, There is an element y in B such that (x,y)∈F

Property 2: For all elements x in A and y and z in B, if (x,y)∈F and (x,z)∈F then y=z

y=F(x) ↔ (x,y)∈F

Inverse Relation

R⁻¹ = {(y,x)∈B X A|(x,y)∈R}

Reflexive

R is reflexive ↔ for all x in A, (x,x)∈R

Symmetric

R is symmetric ↔ for all x and y in A, if (x,y)∈R then (y,x)∈R

Transitive

R is transitive ↔ for all x,y, and z in A, if (x,y)∈R and (y,z)∈R then (x,z)∈R

Equivalence Relations

relations that are reflexive, symmetric, and transitive

[a] = {x∈A|xRa}