Question

problem 19 (contrapositive) consider the following two statements: \100 percent of convicted felons consumed bread during their childhood.\ and \people who do not consume bread as a child do not become felons.\ does the first statement logically imply the second?

Explanation:

Translate the first statement into conditional form

The first statement is: "100 percent of convicted felons consumed bread during their childhood."
This can be written as a conditional statement:
If a person is a convicted felon, then they consumed bread during their childhood.
Let \(P\) be "is a convicted felon" and \(Q\) be "consumed bread during childhood".
The statement is \(P \implies Q\).

Translate the second statement into conditional form

The second statement is: "People who do not consume bread as a child do not become felons."
This can be written as a conditional statement:
If a person did not consume bread during childhood, then they do not become a convicted felon.
Using our symbols, this is \(
eg Q \implies
eg P\).

Analyze the logical relationship

In formal logic, the conditional statement \(P \implies Q\) and its contrapositive \(
eg Q \implies
eg P\) are logically equivalent.
Since the second statement is the exact contrapositive of the first statement, they share the same truth value.
Therefore, the first statement logically implies the second.

Answer:

Yes, the first statement logically implies the second because the second statement is the contrapositive of the first.