discrete - set theory

set

collection of objects; order negligible

empty/null/void set

∅ = {}

integers set

ℤ = {…-2,-1,0,1,2…}

naturals set

ℕ = {0?,1,2…}

rational numbers set

ℚ = {a/b : a, b ∈ ℤ, b ≠ 0}

naturals w/ zero set

ℕ⁰ = ℕ₀ = {0,1,2…}

real numbers set

ℝ = {points on a straight line}

naturals w/o zero set

ℕ⁺ = ℕ₁ = {1,2…}

complex numbers set

ℂ = {a + bi | a, b ∈ ℝ}

∈

element is within; a ∈ A ⇔ A contains a

⊆

subset; A ⊆ B ⇔ all elements of A ∈ B; could be equal

⊂

proper subset; A ⊂ B ⇔ A ⊆ B and A ≠ B

+

positive elements only, excluding 0

nonneg

positive elements only, including 0

-

negative elements only

| |

cardinal; count of unique elements

P()

“power set of X”; all possible subsets of a set // |P(A)| = 2ⁿ

×

cartesian product; all pairs b/w two sets; {(a,b) : a ∈ A, b ∈ B}

∪

union b/w A and B; {x|x ∈ A or x ∈ B}

∩

intersection b/w A and B; {x|x ∈ A and x ∈ B}

A - B

difference b/w A and B; {x|x ∈ A and x ∉ B}

disjoint

A ∩ B = ∅; mutually exclusive; incompatible

pairwise disjoint

collection of disjoint sets

complement

Aᶜ = A' = U - A

associative laws

(A ∪ B) ∪ C = A ∪ (B ∪ C)

(A ∩ B) ∩ C = A ∩ (B ∩ C)

commutative laws

A ∪ B = B ∪ A

A ∩ B = B ∩ A

distributive laws

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

identity laws

A ∪ ∅ = A

A ∩ U = A

complement laws

A ∪ A’ = U

A ∩ A’ = ∅

idempotent laws

A ∪ A = A

A ∩ A = A

bound laws

A ∪ U = U

A ∩ ∅ = ∅

de morgan’s laws

(A ∪ B)’ = A’ ∩ B’

(A ∩ B)’ = A’ ∪ B’