Sets, Logic, and Real Numbers
Core concepts
A candidate must be fluent with set notation: union, intersection, difference, complement, subsets, and the number of subsets of an n-element set (2^n). Know the connectives (negation, conjunction, disjunction, implication, equivalence) and their truth tables, plus quantifiers and their negations. For real numbers, master absolute value as distance, intervals, and inequalities of the form |x − a| < r meaning a − r < x < a + r.
Traps
- Confusing ⊆ with ∈. The empty set is a subset of every set but is an element only of sets that list it.
- Forgetting that the negation of "all x satisfy P" is "some x fails P", not "all x fail P".
- Mishandling strict versus non-strict interval endpoints, and forgetting that dividing an inequality by a negative reverses its direction.
- Assuming √(x²) = x; it equals |x|.
Worked example
Describe the set A = {x ∈ ℝ : |2x − 3| ≤ 5} as an interval. Split: −5 ≤ 2x − 3 ≤ 5, so −2 ≤ 2x ≤ 8, giving −1 ≤ x ≤ 4, i.e. A = [−1, 4]. If B = (0, 6), then A ∩ B = (0, 4] and A ∪ B = [−1, 6).
Rule of thumb
Rewrite absolute-value inequalities as double inequalities before solving, and always check whether endpoints are included by testing the boundary value.