MATH 101 C
First Semester AY 2025-2026
Exercise 2 - Part 1
Perform as indicated. Make sure to provide complete answers written neatly and legibly.
1. The following arguments are valid. Construct a formal proof of validity using the specificied
approach, if there is any.
(a) (Direct Proof)
(c)
(H → I) ∧ (J → K)
(I ∨ K) → L
∼L
∴∼ (H ∨ J)
(∀x), (A(x) → B(x))
(∃x), (A(x) ∨ C(x))
(∀x), (C(x) →∼ D(x))
∴ (∃x), (D(x) → B(x))
(b) (Indirect Proof)
(A ∨ B) → (C ∧ D)
∼ A → (E →∼ E)
∼C
∴∼ E
(d)
(∀x)(A(x) →∼ B(x))
(∀x)(C(x) ∧ A(x))
∴ (∀x)(C(x)∧ ∼ B(x) ∨ A(x))
2. Determine the given argument is valid or not. In either case, use the indirect method to
establish its validity or invalidity.
(a)
(b)
D→E
(D → (D ∧ E)) → (F →∼ G)
G
∴∼ F
(∃x), (A(x) ∨ B(x) → C(x))
(∃x), (A(x) ∧ D(x))
∴ (∀x), (C(x))
3. [5 points] Suppose the following statements are true. Using rules of inference, prove that if
the alarm rings, then the building is safe or the problem is identified.
(a) If the alarm rings, then either the students evacuate or the teacher investigates.
(b) If the students evacuate, then the building is safe.
(c) If the teacher investigates, then the problem is identified.
(d) Either the alarm rings or the system is malfunctioning.
(e) If the building is not safe or the problem is not identified, then either the alarm did not
ring or the system is not malfunctioning.
End of Exercise 2 - Part 1
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MATH 101 C
First Semester AY 2025-2026
Exercise 2 - Part 2
Choose exactly one item from each set. Provide a formal proof of validity for the chosen items.
Make sure to provide complete answers written neatly and legibly.
Set A. (Unique existence proof)
(a) For every integer y, there exists a unique real number x such that 7x + 5y = 35.
(b) There exists a unique integer m such that for every integer n, mn+2m+2n+2 = n.
Set B. (Nonconstructive existence proof)
(a) Let f be a function defined by f (x) = x2 − 4x + 1 on the interval [0, 4]. Then there
exists a real number c on the interval (0, 4) such that f ′ (c) = 0.
(b) Let f be a function defined by f (x) = 2x2 + 5x + 4 on the interval [−1, 3]. Then
there exists a real number c on the interval (−1, 3) such that f ′ (c) = 2.
Set C. (Proof by contraposition)
(a) If n is an integer and n3 + 5 is odd, then n is even.
(b) Suppose a, b, and c are integers. If a2 + b2 = c2 , then a or b is even.
Set D. (Proof by contradiction)
(a) If a and b are integers and a ≥ 2 then a does not divide b or a does not divide
b + 1.
(b) For every real number x on the interval [0, π/2], sin x + cos x ≥ 1.
Set E. (Proof of biconditional statements)
(a) Let a and b be nonzero integers. Prove that a divides b and b divides a if and only
if a = b or a = −b.
x
y
(b) For all nonzero real numbers x and y, we have + ≥ 2 if and only if either
y
x
x > 0 and y > 0 or x < 0 and y < 0.
End of Exercise 2
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