CS 151: Homework 1 Due February 1st, 8:00pm on Gradescope All work must be individual 1 English to Propositional Logic Convert the following English sentences into propositional logic. Define propositions as necessary but make sure that your defined propositions are as simple as possible, i.e. they should not be compound propositions. a) I will be hungry and thirsty if and only if I do not go to dinner b) If it rains outside, I will see you at dinner only if I have time 2 Truth Table Give the truth table for (pq)r and p(qr). Give them as two columns of the same table. 3 Logical Equivalences Prove that the following pairs of compound propositions are equivalent by using the Laws of Propositional Logic. Use one law per line and give a citation. You may use associativity, commutativity or double negation alongside other laws without citation. a) (pq)q and pq b) (pq)(qp) and (qp)(pq) Laws of Propositional Logic You do not need to cite associativity, commutativity or double negation Idempotency Domination - ppp - pTT - ppp - pFF Identity Implication - pFp - pq(pq) - pTp - pq(pq)(qp) DeMorgan's - pqqp - (pq)(pq) Distributivity - (pq)(pq) - p(qr)(pq)(pr) - xP(x)xP(x) - p(qr)(pq)(pr) - xP(x)xP(x) Negation Absorption - ppT - p(pq)p - ppF - p(pq)p Associativity - p(pq)(pq) - p(qr)(pq)r - p(pq)(pq) - p(qr)(pq)r Double Negation Commutativity - pp - (pq)(qp) - (pq)(qp)