Math 240 Midterm Review
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Math 240 Midterm Review
1. (Propositional Logic) Simplify the following statements. Your final answers should use only the symbols p, q, r, ¬, ∨, ∧.
(a) (p ⇒ q) ∧ (q ⇒ ¬p)
(b) (p ⇔ q) ∨ (p ⇔ ¬q)
(c) (p ∨ q) ∧ (p ∨ r) ∧ ¬(q ∧ r)
2. (Quantifiers) Write the negation of each of the following statements. In each case, determine whether the original or the negation is true.
(a) (In the universe of P(N)) ∀X∃Y ( (X ∩ Y = ∅) ∧ (X ∪ Y = N))
(b) (In the universe of Z) ∀x∃y∀z(yz2 < x)
(c) (In the universe of Z) ∀x∃y( (x < 0) ∨ (x = y2))
3. (Sets) Let A, B be sets.
(a) Prove that (A ∩ B) × B ⊆ B × B.
(b) Prove that A × B ⊆ A × (A ∪ B).
(c) Suppose that A ∪ B = B. Which of the following is true:
i. A = B.
ii. A ⊆ B.
iii. B ⊆ A.
4. (Functions) Determine whether the following functions are injective/surjective/bijective.
(a) f : Z × Z → Z defined by
f(a, b) = ab
(b) f : R → Z defined by
f(x) = x
Recall the floor function x which gives the largest integer less than or equal to x. For example: 2.9 = 2, π = 3, −5.5 = −6.
(c) f : P(N) × P(N) → P(N) defined by
f(X, Y) = X ∩ Y
5. (Proofs)
(a) Prove the following statement: If a and b are rational then a + b is rational.
(b) Prove the following statement: If a is rational and b is irrational then a + b is irrational.
(c) Consider the similar statement: If a and b are irrational then a + b is irrational. Is this statement true?
(d) Prove the following statement: If a is rational then a2 + a + 1 0.
6. (Relations) For each of the following relations, determine whether the relation is reflexive/symmetric/transitive (assume the universe is R).
(a) aRb if a b.
(b) aRb if ab 0.
(c) aRb if 5|(b − a).
(d) aRb if a = b + 1.
(e) aRb if a + b = 0.
2021-10-19