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
(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) βΆ 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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