Question

If the statement $p \to \left( {q \vee r} \right)$ is true then the truth values of statements p, q, r respectively.
A) T, F, T
B) F, T, F
C) F, F, F
D) All of these

Answer 1

Hint:
In this case $p \to \left( {q \vee r} \right)$ is true so p and (qvr) is true. Form three cases taking q and r true and false alternatively and both true, then find truth values for each case.

Complete step by step solution:
Truth-value, in logic, truth (T or 1) or falsity (F or O) at a given proposition or statement.
$p \to \left( {q \vee r} \right)$mean P implies (q or r)
And $p \to \left( {q \vee r} \right)$ is true means P is true and (qvr) is also true.
so, the truth value of P is T.
Now (qvr) is also true.
It is true in three cases.
Case-1. if q is true and r is false.
 $q \to T,\;r \to F$
Truth values of P, q, r is (T,T,F)
Case-2. if q is false and r is true.
$q \to F,\;r \to T$
Truth values of P,q,r is (T,F,T)
Case-3. if q and r are true.
 $q \to T,\;r \to T$
Truth value of p,q,r is (T,T,T)
Now from the options we can see that only option (A) matches our result.

So, option (A) is the correct answer.

Note:
Meaning of other symbols:-
 $p \vee q \to p\;or\;q$
$p \wedge q \to p\;and\;q\;$
P → negation of P etc.
This question can be asked using both ‘and’ and ‘or’.