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This is a midterm exam for cs 173, a course in discrete mathematics and formal systems, covering topics such as logic, set theory, and mathematical induction. The exam consists of multiple choice, short answer, and proof-based problems.
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Which of the following is logically equivalent to p → q?
a) q ∨ ¬p
b) the contrapositive of p → q
c) the inverse of the converse of p → q
d) all of the above
Which of the following is a negation of ∀x∀y[((x > 0) ∧ (y > 0)) → (x + y > 0)]?
a) ∃x∃y[(x > 0) ∧ (y > 0) ∧ (x + y ≤ 0)]
b) ∃x∃y[((x ≤ 0) ∨ (y ≤ 0)) ∧ (x + y > 0)]
c) ∀x∀y¬[((x > 0) ∧ (y > 0)) → (x + y > 0)]
d) ∃x∃y[(x ≤ 0) ∨ (y ≤ 0) ∨ (x + y > 0)]
Let f (x) = 3x + 2 and g(x) = x^2 be functions defined on the integers (f : Z → Z, g : Z → Z). Which of the following is true?
a) g ◦ f = O(x^2 )
b) g ◦ f = O(x^3 ), and g ◦ f is not O(x^2 )
c) g ◦ f (x) = f ◦ g(x)
d) g ◦ f has an inverse function.
Which of the following is false?
a) {x} ⊆ {x}
b) {x} ∈ {x, {x}}
c) {x} ⊆ P({x}), where P({x}) is the power set of {x}
d) {x} ⊆ {x, {x}}
Use an indirect proof for the following:
Given: p → (m → w) w → d m ¬d Prove: ¬p
Tell whether each of the following is True or False. The universe is all integers.
a) ∀z∀y∃x(x − y = z)
b) ∀y∃x∃z(x − y = z)
c) ∀x∀y∀z(x − y = z)
d) ∀x∀y∃z(x − y = z)
e) ∀x∃y∃z(x − y = z)
f) ∃x∃y∀z(x − y = z)
g) ∃x∃y∃z(x − y = z)
h) ∃x∀y∀z(x − y = z)
Prove that 8 n^2 + n is O( n
2 2 −^ 5).
