Sets and logic · Grade 10
In Grade 10 you learn: sets and notation.
Sets collect objects together. Logic makes precise what “and”, “or”, “not” and “if … then” mean. Together they are the grammar of mathematics. They also power database searches and computer circuits.
Key facts
- A ∪ B (union): in A or B or both. A ∩ B (intersection): in both. A′ (complement): not in A. ∅ is the empty set.
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B). For three sets: add the singles, subtract the pairs, add back the triple.
- A set with n elements has 2ⁿ subsets, and n(A × B) = n(A) × n(B).
- ¬ means not, ∧ means and, ∨ means or, and ⇔ means if and only if. ⇒ means implies: it is false only when a true statement implies a false one.
- De Morgan’s laws: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q and ¬(P ∨ Q) ≡ ¬P ∧ ¬Q.
- P ⇒ Q is equivalent to its contrapositive ¬Q ⇒ ¬P, but not to its converse Q ⇒ P.