The Gateway to Computer Science Excellence
+1 vote
1.3k views
57. Let P, Q, R and S be Propositions. Assume that the equivalences P ⇔ (Q ∨ ¬ Q) and Q ⇔ R hold. Then the truth value of the formula (P ∧ Q) ⇒ ((P ∧ R) ∨ S) is always :

(1) True

(2) False

(3) Same as truth table of Q

(4) Same as truth table of S
in Discrete Mathematics by (399 points)
recategorized by | 1.3k views

1 Answer

+5 votes
Best answer

P ⇔ (Q ∨ ¬ Q) "P should be true because RHS will be TRUE always "

Q ⇔ R "when Q is true R is true" and  "when Q is false R is false"

 $(P ∧ Q) ⇒ ((P ∧ R) ∨ S)$

there can be only 2 cases (value of S doesn't matter)

1) P = True, Q = True and R = True

        $(T ∧ T) ⇒ ((T ∧ T) ∨ S)$ 

         so this case is True

2) P =True, Q = R = False

        $(T ∧ F) ⇒ ((P ∧ R) ∨ S)$

        In case implication if the premises is false then whole statement is true

        this case is also true

The given expression is True in both the cases.

Answer is 1) True

by Boss (36.4k points)
selected by
0
What if R is false in your ist point ?

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
50,737 questions
57,257 answers
198,086 comments
104,735 users