Implication, predicates and quantifiers
Answers are for parents and teachers.
1
For “if a triangle is equilateral, all its angles are equal” write the converse, inverse and contrapositive and say whether each is true.
Converse: if all angles are equal, it is equilateral — true. Inverse: if it is not equilateral, its angles are not all equal — true. Contrapositive: if the angles are not all equal, it is not equilateral — true.
2
p: “3² = 6”, q: “4 > 1”. Find the values of p ⇒ q, q ⇒ p, p ⇔ q.
p false, q true: p ⇒ q true, q ⇒ p false, p ⇔ q false.
3
Write the truth table of (p ⇒ q) ∧ (q ⇒ p) for (p, q) = TT, TF, FT, FF. Which statement does it match?
T, F, F, T — this is p ⇔ q.
4
D = {1, 2, 3, 4, 5, 6}, P(x): x² − 7x + 10 ≤ 0. Find the values of ∀xP(x) and ∃xP(x); give a counterexample.
P(x) holds for x = 2, 3, 4, 5 (x² − 7x + 10 = (x − 2)(x − 5) ≤ 0). ∃xP(x) is true, ∀xP(x) is false; counterexample x = 1 (1 − 7 + 10 = 4 > 0).
5
Write the negation of “every student of 10-A solved at least one problem”.
There is a student in 10-A who solved no problem at all.