Answer to Question #238733 in Discrete Mathematics for amanuel

Question

If the truth value of ( p ➡️ q) v negation r is false, then what is the truth value negation q ↔️r ?

Answer 1

If I understood task correctly, here is the condition:

Given

p -> q V !r = False

What is the value of

!(q <-> r)

?

Solution:

1. Let's simplify !(q <-> r) expression:

!(q <-> r)
!(q -> r ^ r -> q)
!(q -> r) V !(r -> q)
!(!q V r) V !(!r V q)
(q ^ !r) V (r ^ !q)

2.Let's find out truth values of p,q and r (table below). We see that

p -> q V !r = False

only when p = True, q = False and r = True.

| p | q | r | p -> q | p -> q V !r |

| F | F | F | T | T |

| F | F | T | T | T |

| F | T | F | T | T |

| F | T | T | T | T |

| T | F | F | F | T |

| T | F | T | F | F |

| T | T | F | T | T |

| T | T | T | T | T |

1. Let's substitute truth values for p, q and r to find truth value of

(q ^ !r) V (r ^ !q)

which is simplified version of

!(q <-> r)

(q ^ !r) V (r ^ !q)
(F ^ F) V (T ^ T)
F V T
True

Thus, truth value of

!(q <-> r)

is True.