Question

Determine whether the biconditional statement is true or false. If false, give a counterexample.

The number k is a positive integer if and only if 5k is a natural number.


A. true
B. false; 5k = 1, then k = −1/5, which is not a positive integer
C. false; 5k = 25, then k = 5 which is not a positive integer
D. false; 5k = 1, then k = 1/5, which is not a positive integer

Answer 1

A biconditional statement is something of the form:

P if and only if Q.

For two given propositions P and Q.

We will see that the correct option is D: " false; 5k = 1, then k = 1/5, which is not a positive integer"

The biconditional statement: P if and only if Q.

implies that:

If P is true, then Q is true.

If Q is true, then P is true.

if Q is false, then P is false

if P is false, then Q is false.

Here the statement is:

"The number k is a positive integer if and only if 5k is a natural number"

Now, notice that "5*k is a natural number" can be true if:

k = (1/5)

So we get: 5*(1/5) = 1 is a natural number.

So 5k can be a natural number in cases where k is not a positive integer (for example, or k = 1/5, 2/5, etc...)

So we just found a counterexample of the statement, thus the statement is false, and the correct option is D "false; 5k = 1, then k = 1/5, which is not a positive integer"

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