Sets and Common Logical Connectives
Sets and common logical connectives
A set is fixed by its elements; master union, intersection, set difference, complement relative to a universal set, subsets, and cardinality for finite sets, including |A∪B|=|A|+|B|−|A∩B|. Remember that ∅⊆A and A⊆A always hold, and distinguish an element x∈A from a subset {x}⊆A. In logic, distinguish inclusive “or” from exclusive “or,” and translate “if P then Q” as P⇒Q, whose contrapositive ¬Q⇒¬P is equivalent. “P is necessary for Q” means Q⇒P; “P is sufficient for Q” means P⇒Q. Quantified statements negate by swapping ∀ and ∃ and negating the predicate: ¬(∀x,P(x)) is ∃x,¬P(x). De Morgan's laws apply to sets and propositions.
Common traps: overlooking the empty set as a subset; mishandling endpoints in intervals; treating the converse or inverse as equivalent to the original; and forgetting that A⊆B fails if one element of A lies outside B. When a parameter appears, test boundary values separately.
Worked rule: solve A={x:x²−3x+2≤0}=[1,2]. For B={x:x<a}, A⊆B requires every x in [1,2] to satisfy x<a, so a>2, not a≥2. At a=2, the point x=2 belongs to A but not B. Number-line sketches with open and closed endpoints prevent this error. For quantified logic, rewrite the negation before judging truth.