Download Review of logic and proofs

Review: logic and proofs
MATH 174
1. Give the truth tables for (a) ¬p, (b) p ∨ q, (c) p ∧ q, (d) p → q, and (e) p ↔ q.
2. Give the truth table for (p → q) ∨ (r ↔ q).
3. State the following equivalences: (a) DeMorgan’s Law (both forms), (b) turning a conditional into a disjunction, (c) turning a false conditional into a conjunction, (d) turning a
biconditional into two conditionals.
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4. Using the equivalences from (3), together with distribution, associativity, commutativity,
show that
(p → q) ∨ (r ↔ q) ≡ (q ∧ ¬p) → [(r → q) ∧ (q → r)].
5. Devise your own translation scheme and symbolize the following sentences. Our universe
is all animals (including people).
(a) Socrates is a person.
(b) All people are mortal.
(c) There is no mortal person.
(d) Socrates gets eaten by a fish.
(e) All mortal people get eaten by a fish.
(f) There exists a mortal animal that gets eaten by a fish but is not a person.
6. Let P (x) = “x is a math major,” Q(x) = “x is taking MATH 174,” and R(x, y) = “x
lives with y.” Let a = Alice, b = Bob, c = Cat, and d = Derrick. Write the following
sentences in the simplest English possible.
(a) ∀x[P (x) → Q(x)]
(b) ∃xy[P (x) ∧ ¬P (y) ∧ R(x, y)]
(c) ∀x[P (x) ∧ ∃yR(x, y) → P (y)]
(d) [P (d) ∧ (P (d) → Q(d))] → Q(d)
(e) [(Q(b) ∨ R(a, c)) ∧ ¬Q(b)] → ¬R(a, c)
(f) ∀xP (x) → ∃xP (x)
7. Of (6d)-(6f) above, which statements are true?
8. The following are attempts to prove that n being an integer and n2 being even implies
that n is even. Find the mistake.
(a) Theorem. If n is an integer and n2 is even, then n is even. Proof. Let n be an integer
and n2 be even.√Then we may write n2 = 2m for some integer m and take the square root,
giving us n = 2 m. Therefore, n is even.
(b) Theorem. If n is an integer and n2 is even, then n is even. Proof. Suppose n is an
integer and n2 is even. Next, assume n is even. We may thus write n = 2m for some integer
m. Then,
n2 = (2m)2 = 2(2m2 ).
Since n2 is even, this completes the proof.
9. Either prove or disprove (find a counterexample to) each of the following. If you prove a
theorem, do so without appealing to any theorems from class (i.e., you may have to repeat
some work).
(a) If n is an integer, then n is odd if and only if its square is.
(b) The sum of two rational numbers is rational.
(c) The sum of two irrational numbers is irrational.
(d) The square root of 2 is irrational.