**Question 1**

- A set of premises is given below

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(π βΆ π) β¨ (~π)

(~π) β§ (π β¨ π)

π βΆ (~π)

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.

(~ π) β¨ (~π)

π βΆ (π β§ (~π))

π βΆ (π β§ ~π)

- Construct a chain of logical equivalences to show that

(~π β§ π) βΆ (π βΆ π) β‘ (~π) βΆ (π βΆ ~π).

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.

π β¨ π

(π β§ π) βΆΒ *s ~ s β΄ π βΆ π *

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.

βπ₯ β β βπ¦ β β βπ§ β β ((π₯2 < π¦2) β¨ (π¦2 < π§2)) βΆ ((π₯ < π¦) β¨ (π¦ < π§))

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

βπ₯ β β€ βπ¦ β β βπ§ β β ((π₯ β 0) β§ (π₯π¦)π§ = 1) βΆ ((π§ = 0) β¨ (π₯π¦ = 1))

- Let π· be the set

π· = {β10, β9, β7, β6, β4, β3, β2,0,1,2,3,4,5,6,9,10,12,13,14}.

Suppose that the domain of the variable π₯ is π·. Write down the truth set of the predicate.

((π₯ > 1) βΆ (π₯ is even)) βΆ (π₯ is divisible by 4).

- Let π,π, π , π denote predicates. Use the Rules of Inference to prove that the following argument form is valid.

βπ₯ (π(π₯) βΆ (βπ¦ π(π¦)))

βπ₯ (π
(π₯) βΆ (βπ¦ ~π(π¦)))

βπ₯ (π
(π₯) β§ π(π₯))

β΄ βπ₯ ~π(π₯

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