Relations and Functions

$A × B = {(a,b) : a ∈ A, b ∈ B}; n(A×B) = n(A)·n(B)$

Set of ALL ordered pairs with first element from A and second from B.

$2^(n(A)·n(B))$

Total number of subsets of A×B; each subset is a valid relation.

$I(x) = x, ∀ x ∈ domain$

Maps every element to itself; graph is $y = x$ line through origin.

$f(x) = |x| = x if x ≥ 0; −x if x < 0$

$V-shaped$ graph; domain ℝ, range $[0,∞).$

$⌊x⌋ = greatest integer n such that n ≤ x$

Step function; constant on $[n, n+1),$ jumps at integers.

$sgn(x) = 1 if x>0; 0 if x=0; −1 if x<0$

Returns only the sign of x. Alternative: $sgn(x) = x/|x|$ for $x≠0.$

$Onto functions from an m-set to an n-set = Σₖ₌₀ⁿ (−1)ᵏ ⁿCₖ (n−k)ᵐ$

$Inclusion-exclusion$ over the codomain: subtract functions that MISS at least one codomain element.

$f(x) = fₑ(x) + f₀(x), fₑ(x)=(f(x)+f(−x))/2 (even part), f₀(x)=(f(x)−f(−x))/2 (odd part)$

Every function with a domain symmetric about $0$ splits uniquely into an even part and an odd part.

$If f has period T₁ and g has period T₂, then f+g (or fg) has period LCM(T₁,T₂) — PROVIDED this LCM is actually attained$

The LCM rule gives only an UPPER BOUND on the period; the true fundamental period can be smaller.