2 Determine whether each of the conditional statements is tr

2. Determine whether each of the conditional statements is true or false.

Justify your response.

(a) (4 points) 2 + 2 = 4 if and only if 1 + 1 = 3.

(b) (4 points) 2 + 2 = 4 only if 1 + 1 = 3.

(c) (4 points) If 2 + 2 = 4, then 1 + 1 = 3.

Solution

(a) Consider the given bi conditional statement

    \" 2 + 2 = 4 if and only if 1 + 1 = 3 \"

Let us denote

P : 2 + 2 = 4    and    Q : 1 + 1 = 3

Then The truth value of P is T(True) and the truth value of Q is F(False)

But the biconditional statement PQ has the truth value T if both P and Q are True or both are False otherwise P Q has the truth value F.

Hence the given statement \" 2 + 2 = 4 if and only if 1 + 1 = 3 \" has the truth value False.

(b) Consider the given conditional statement
    \" 2 + 2 = 4 only if 1 + 1 = 3\"

Then denote P : 1 + 1 = 3   and Q : 2 + 2 = 4

Since P has the truth value False and Q has the truth value True.

Then the given conditional statement P Q has the truth value True.

(c) Consider the given conditional statement
    \" If 2 + 2 = 4 then 1 + 1 = 3 \"

Let us denote

P : 2 + 2 = 4    and    Q : 1 + 1 = 3

Then The truth value of P is T(True) and the truth value of Q is F(False)

Then the given conditional statement P Q has the truth value False.