Logical Reasoning
What you must understand
Logical reasoning on this exam rests on sets, Venn diagrams, and conditional (if–then) statements. Know the vocabulary of set theory: universal set, subset, complement, intersection, union, and disjoint sets. Be able to translate a word problem into a two- or three-circle Venn diagram, fill the innermost region first, and work outward using the totals.
For conditionals, know the parts: hypothesis (the "if" clause) and conclusion (the "then" clause). The converse swaps them, the inverse negates both, and the contrapositive swaps and negates both. Only the contrapositive is logically equivalent to the original statement. A single counterexample disproves a universal claim; a valid deduction must follow necessarily from the premises.
Traps to watch for
- Confusing "and" (intersection) with "or" (union). "Or" in mathematics is inclusive unless stated otherwise.
- Forgetting to subtract the overlap when adding two set counts.
- Treating the converse as equivalent to the original.
- Misreading "some" or "all" statements; "some A are B" does not imply "some A are not B."
- In Venn problems, placing a number in a region before accounting for the overlap.
Rule of thumb
Inclusion–exclusion: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
Worked example: In a class of 30, 18 play soccer, 15 play basketball, and 6 play both. Then n(A ∪ B) = 18 + 15 − 6 = 27, so 30 − 27 = 3 play neither.