Introduction to Sentential Logic Part 2 Solutions

Author: Indigo Curnick

Date: 2025-04-19

You can find the relevant blog here

1. \(P(J) \wedge P(B)\) where \(P(X)\) means "\(X\) will come to the party", \(J\) means "John" and \(B\) means "Barry"
2. \(G(L) \vee G(B)\) where \(G(X)\) means "I will go to \(X\)", \(L\) means "London" and \(B\) means "Bristol"
3. \(T(R) \vee T(S) \wedge \neg (T(R) \wedge T(S))\), where \(T(X)\) means "it will \(X\) tomorrow", \(R\) means "rains" and \(S\) means "snows" (we could alternatively represent this as \(T(R) \text{ xor } T(S)\) if we allow for an \(\text{xor}\) operator)
4. \(P(J) \wedge P(B) \vee (\neg P(J) \wedge \neg P(B))\) but we can also use DeMorgan's law to make this \(P(J) \wedge P(B) \vee \neg (P(J) \vee P(B))\). \(P(X)\) means "\(X\) will come to the party", \(J\) means "John" and \(B\) means "Barry"
5. \(\neg ((T(J) \wedge M(J)) \wedge (T(B) \wedge M(B)))\) but we can also use deMorgan's law to turn this into \(\neg (T(J) \wedge M(J)) \vee \neg (T(B) \wedge M(B))\). \(T(X)\) means "\(X\) is tall", \(M(X)\) means "\(X\) is muscular", \(J\) means "John" and \(B\) means "Barry"

1. \(P \vee Q\)
2. \(P\) (if this one stumps you, try making a truth table for \(Q \vee \neg Q\))
3. \(P \wedge \neg (Q \wedge R)\)
4. \(P \vee (\neg R \wedge Q)\)