ConceptSets & Inequalities

Key formulas

Union: A ∪ B = { x : x ∈ A or x ∈ B }
Intersection: A ∩ B = { x : x ∈ A and x ∈ B }
Complement (relative to universe U): A^c = { x ∈ U : x ∉ A }
Difference: A − B = { x : x ∈ A and x ∉ B }
Inclusion-exclusion: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Subset count of an n-element set: 2ⁿ subsets; 2ⁿ − 1 proper subsets; 2ⁿ − 1 nonempty subsets
Empty set ∅ is a subset of every set
Quadratic inequality ax²+bx+c>0: solution keyed to root(s) and sign of a / discriminant Δ=b²−4ac (Δ>0 two roots, Δ=0 one, Δ<0 none)
Roots straddle k (one >k, one <k) for a>0: f(k) < 0
Rational inequality f(x)/g(x) ≤ 0: sign chart on numerator/denominator zeros (g(x) ≠ 0)
Absolute value: |x − a| < b ⇔ a − b < x < a + b (b>0)
Inequality multiplicative property: multiplying/dividing by a negative reverses the inequality

If the concept underneath still feels loose, patch it here before returning to CSCA questions.

Properties of inequalities

symmetry, transitivity, additive, multiplicative (sign-flip on negative multiply)

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