Given p is false, q is true,and r is false, find the truth value of the statement. (~P <____> ~Q) ____> ~r
Show step by step work.
The <-> operator is true only if both operands have the same truth value, i.e. TT or FF.
Given p is false and q is true, then ~p is true and ~q is false, therefore ~p <-> ~q is false.
Since the conditional p->q is false only if p is true and q is false, here we have
F->~r or
F->T
so it is true. Thus with p=F, q=T and r=F, the whole statement together is true,
To find the truth value of the statement (~P <____> ~Q) ____> ~r, we can break it down into smaller steps.
Step 1: Evaluate ~P
Since we are given that p is false, the negation of false is true. Therefore, ~P is true.
Step 2: Evaluate ~Q
Since we are given that q is true, the negation of true is false. Therefore, ~Q is false.
Step 3: Evaluate ~P <____> ~Q
The symbol <____> represents the biconditional operator, which means "if and only if." It is true when both sides have the same truth value, and false otherwise.
In this case, ~P is true and ~Q is false. Since they have different truth values, the biconditional operator evaluates to false.
Step 4: Evaluate (~P <____> ~Q) ____> ~r
The symbol ____> represents the conditional operator, which means "implies." It is false only when the antecedent (the part before the arrow) is true and the consequent (the part after the arrow) is false. Otherwise, it is true.
In this case, (~P <____> ~Q) is false and ~r is false. Since both the antecedent and the consequent are false, the conditional statement evaluates to true.
Therefore, the truth value of the statement (~P <____> ~Q) ____> ~r is true.