9.2 Coursework Sheet 1
Hand in your solutions in the appropriate way by Monday, 16 October 2023, 19:00.
Exercise 1
Let \(P, Q\) and \(R\) be logical statements. Show that
- \((P\land\lnot Q)\implies R\) is logically equivalent to \(P\implies (Q\lor R)\).
- \((P\land Q)\implies R\) is logically equivalent to \(P\implies (Q\implies R)\).
- \(P\land(Q\lor R)\) is logically equivalent to \((P\land Q)\lor (P\land R)\).
- \(P\lor(Q\land R)\) is logically equivalent to \((P\lor Q)\land (P\lor R)\).
- \((P\implies Q)\land (P\implies R)\) is logically equivalent to \(P\implies(Q\land R)\).
- \((Q\implies P)\land (R\implies P)\) is logically equivalent to \((Q\lor R)\implies P\).
- \((\lnot P)\Leftrightarrow Q\) is logically equivalent to \(P\Leftrightarrow(\lnot Q)\).
Exercise 2
Using truth tables, demonstrate the following `rules of logic’
- Double negation rule. \(P\) is logically equivalent to \(\lnot\lnot P\).
- De Morgan’s rule. \(\lnot(P\lor Q)\) is logically equivalent to \((\lnot P)\land(\lnot Q)\). \(\lnot(P\land Q)\) is logically equivalent to \((\lnot P)\lor(\lnot Q)\).
- Implication rule. \(P\implies Q\) is logically equivalent to \((\lnot P)\lor Q\).
- Contrapositive rule. \(P\implies Q\) is logically equivalent to \((\lnot Q)\implies (\lnot P)\).
Exercise 3
Let \(P, Q\) and \(R\) be logical statements. State the negation, converse and contrapositive of the following statements.
- \((P\lor Q)\implies \lnot R\).
- \(P\implies (Q\implies R)\).
- \((P\implies Q)\implies Q\).
- \(P\implies (Q\land\lnot Q)\).
Exercise 4
Let \(P, Q\) and \(R\) be logical statements. Show that the following statements are tautologies.
- \((\lnot P)\implies(P\implies Q)\).
- \(((\lnot Q)\land (P\implies Q))\implies\lnot P\).
- \(((\lnot P)\land(P\lor Q))\implies Q\).