Week 2 exercise - solution PDF

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Which of the following sentences are propositions? What are the truth values of those that are propositions? (a) Austin is the capital of Texas. (b) Addis Ababa is the capital of Egypt. (c) 2 + 4 = 6 (d) X + 3 = 12 2) A proposition is a statement which can either be true or false (a) Proposition, Tr...

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1) Which of the following sentences are propositions? What are the truth values of those that are propositions? (a) Austin is the capital of Texas. (b) Addis Ababa is the capital of Egypt. (c) 2 + 4 = 6 (d) X + 3 = 12 2)

A proposition is a statement which can either be true or false (a) Proposition, True truth value (or True) (b) Proposition, False truth value (or False) (c) Proposition, True truth value (or True) (d) Not a proposition.

What is the negation of each of these propositions? (a) Charles has a smartphone. (b) There is no pollution in London. (c) 3 + 2 = 5 (d) The summer in Scotland is hot and sunny. (e) Amy has more than 240 GB free hard disk space on her laptop.

Negation is the opposition of a proposition (a) Charles does NOT have a smartphone (b) There is pollution in London (c) 3 + 2 ≠ 5 (d) The summer in Scotland is NOT (hot and sunny). => The summer in Scotland is NOT hot or it is NOT sunny. Note: De Morgan's laws:  

not (A or B) = not A and not B not (A and B) = not A or not B

(e) Amy does NOT have more than 240 GB free hard disk space on her laptop. (Or Amy has equal or less than 240 GB free hard disk space on her laptop).

3) Let p and q be the following propositions respectively: p: Swimming at the shore is allowed q: Sharks have been spotted near the shore Express each of the following compound propositions as an English sentence. (a) ¬q Sharks have NOT been spotted near the shore

(b) p  q Swimming at the shore is allowed AND Sharks have been spotted near the shore

(c) ¬p ∨ q Swimming at the shore is NOT allowed OR sharks have been spotted near the shore

(d) p → ¬q IF swimming at the shore is allowed, THEN sharks have NOT been spotted near the shore. (e) ¬q → p IF sharks have NOT been spotted near the shore, THEN swimming at the shore is allowed.

(f) ¬p → ¬q IF swimming at the shore is NOT allowed, THEN sharks have NOT been spotted near the shore.

(g) p↔ ¬q Swimming at the shore is allowed, IF AND ONLY IF sharks have NOT been spotted near the shore.

(h) ¬p  (p ∨ ¬q) Swimming at the shore is NOT allowed AND, either swimming at the shore is allowed or sharks have NOT been spotted near the shore. Note: we can incorporate the parenthesis by using the word ‘either’...