University |
Singapore University of Social Science (SUSS) |

Subject |
MTH105: Fundamentals of Mathematics |

^{th}Aug 2023

# MTH105: Fundamentals of Mathematics Assignment, SUSS, Singapore A set of premises is given below. Determine which of the following statements is a valid conclusion

**Question 1**

- A set of premises is given below

(𝑝 ⟶ 𝑞) ∨ (~𝑟)

(~𝑝) ∧ (𝑞 ∨ 𝑟)

𝑟 ⟶ (~𝑝)

Determine which of the following statements is a valid conclusion from the above set of premises using truth tables or by providing a logical explanation.

(~ 𝑝) ∨ (~𝑟)

𝑞 ⟶ (𝑝 ∧ (~𝑞))

𝑝 ⟶ (𝑞 ∧ ~𝑟)

- Construct a chain of logical equivalences to show that

(~𝑞 ∧ 𝑟) ⟶ (𝑝 ⟶ 𝑞) ≡ (~𝑞) ⟶ (𝑝 ⟶ ~𝑟).

Do not use truth tables in this part of the question.

- Use the Rules of Inference to prove that the following argument form is valid.

𝑞 ∨ 𝑟

(𝑝 ∧ 𝑞) ⟶ *s
~ s
∴ 𝑝 ⟶ 𝑟
*

Do not use truth tables in this part of the question.

**Question 2**

- Give a counter-example to show that the following statement is false.

∀𝑥 ∈ ℕ ∀𝑦 ∈ ℝ ∀𝑧 ∈ ℝ ((𝑥2 < 𝑦2) ∨ (𝑦2 < 𝑧2)) ⟶ ((𝑥 < 𝑦) ∨ (𝑦 < 𝑧))

- Provide the negation of the statement, giving your answer without using any logical negation symbol. Equality and inequality symbols such as =, ≠, <, > are allowed.

∃𝑥 ∈ ℤ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥 ≠ 0) ∧ (𝑥𝑦)𝑧 = 1) ⟶ ((𝑧 = 0) ∨ (𝑥𝑦 = 1))

- Let 𝐷 be the set

𝐷 = {−10, −9, −7, −6, −4, −3, −2,0,1,2,3,4,5,6,9,10,12,13,14}.

Suppose that the domain of the variable 𝑥 is 𝐷. Write down the truth set of the predicate.

((𝑥 > 1) ⟶ (𝑥 is even)) ⟶ (𝑥 is divisible by 4).

- Let 𝑃,𝑄, 𝑅, 𝑆 denote predicates. Use the Rules of Inference to prove that the following argument form is valid.

∀𝑥 (𝑃(𝑥) ⟶ (∀𝑦 𝑄(𝑦)))

∀𝑥 (𝑅(𝑥) ⟶ (∃𝑦 ~𝑄(𝑦)))

∃𝑥 (𝑅(𝑥) ∧ 𝑆(𝑥))

∴ ∀𝑥 ~𝑃(𝑥)

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