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.

Parameters from set conditions (equality/complement/intersection) with element-distinctness constraints

No misconceptions are logged for this concept yet, so there is nothing named here to trip you up. Practice it and anything that does catch you will be added.

Practice this concept →