Computational Mathmatics

Victorian Institute of Technology Pty Ltd

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ITSU2011 – Computational Mathematics Page 1 of 3

ITSU2011

Computational Mathematics

Activity 02

Victorian Institute of Technology Pty Ltd

ABN: 41 085 128 525 RTO No: 20829 TEQSA ID: PRV14007 CRICOS Provider Code: 02044E

ITSU2011 – Computational Mathematics Page 2 of 3

Q1) Translate the following into English.

A. ∀𝑥 𝑀𝐴𝑁(𝑥) where 𝑥 is a man.

B. ∀𝑥 𝐶𝐻𝐴𝑆𝐸(𝑥, 𝑠) where 𝑠 is Sam.

C. ∀𝑥 ∀𝑦 𝐿𝑂𝑉𝐸(𝑥, 𝑦) where 𝐿𝑂𝑉𝐸(𝑥, 𝑦) denotes 𝑥 loves 𝑦

D. ∀𝑥 ∃𝑦 𝐿𝑂𝑉𝐸(𝑥, 𝑦)

E. ∃𝑦 ∀𝑥 𝐿𝑂𝑉𝐸(𝑥, 𝑦)

F. ∃𝑥 (𝑀𝐴𝑁(𝑥) ∧ 𝑥 = 𝑎) where 𝑥 is man and 𝑎 is Anthony.

Q2) Let 𝑃(𝑥) = person, 𝑆(𝑥) = is sleeping, translate the following to predicate logic.

A. Someone is sleeping.

B. No one is sleeping

C. Everyone is sleeping

D. Not everyone is sleeping

E. Everyone is sleeping

Q3) Let 𝐷(𝑥) = dragon, 𝐺(𝑥) = green, 𝑆(𝑥) = is sleeping, translate the following to predicate

logic.

A. Some green dragon is sleeping

B. No green dragon is sleeping

C. Every green dragon is sleeping

D. Not every green dragon is sleeping

Q4) Let 𝐷(𝑥) = dragon, 𝑇(𝑥) = twitching, 𝑆(𝑥) = is sleeping, translate the following to

predicate logic.

A. Some dragon is sleeping or twitching

B. No dragon is sleeping or twitching

C. Every dragon is sleeping or twitching

D. Not every dragon is sleeping or twitching

Q5) Prove that the sum of the first n natural numbers is given by this formula:

1 + 2 + 3 + . . . + 𝑛 = 𝑛(𝑛+1)

2 for all 𝑛 ≥ 1

Victorian Institute of Technology Pty Ltd

ABN: 41 085 128 525 RTO No: 20829 TEQSA ID: PRV14007 CRICOS Provider Code: 02044E

ITSU2011 – Computational Mathematics Page 3 of 3

Q6) Prove by mathematical induction that 𝑛! + (𝑛 − 1)! + (𝑛 − 2)! = 𝑛2(𝑛 − 2)! for all 𝑛 ≥ 2