Question

If I mow my neighbor's lawn, then I will earn $5. Which of the following is a logically equivalent statement to the conditional statement above?

a) If I earn $5, then I mowed my neighbor's lawn.
b) If I don't mow my neighbor's lawn, then I won't earn $5.
c) If I don't earn $5, then I didn't mow my neighbor's lawn.
d) If I earn $5, then I didn't mow my neighbor's lawn.

Answer 1

The logically equivalent statement to the conditional 'If I mow my neighbor's lawn, then I will earn $5' is 'If I don't earn $5, then I didn't mow my neighbor's lawn', which is option (c). This is because the contrapositive of a conditional statement is always logically equivalent to the original.

Step-by-step explanation:

The conditional statement 'If I mow my neighbor's lawn, then I will earn $5' can be analysed in terms of logic to find a statement that is logically equivalent. In logic, the contrapositive of a conditional statement is always logically equivalent to the original statement. The contrapositive is formed by negating both the antecedent and the consequent of the original statement and then reversing them. Therefore, the logically equivalent contrapositive statement to 'If I mow my neighbor's lawn, then I will earn $5' is 'If I don't earn $5, then I didn't mow my neighbor's lawn'.

Thus, the answer is option (c) If I don't earn $5, then I didn't mow my neighbor's lawn. It is important to note that the inverse or the converse of the original statement are not logically equivalent to the conditional statement, making the other options incorrect.