Question 28

If the children are not playing outside, then the television is off.

This statement is logically equivalent to the original statement because it presents the contrapositive of the initial condition. The contrapositive of "If the television is off, then the children are playing outside" is "If the children are not playing outside, then the television is not off," which maintains truth in logical equivalence.

A) If the television is not off then the children are not playing outside

This option suggests a direct relationship that is not supported by the original statement. The original does not imply that the television being on directly causes the children to not play outside, making this statement incorrect.

B) If the children are not playing outside then the television is off

This option is correct as it is the contrapositive of the original statement. In logical terms, a statement is equivalent to its contrapositive, thus affirming the truth of the original condition in the inverse scenario.

C) If the children are playing outside then the television is off

This option incorrectly reverses the original statement's implication. While it addresses the scenario of the children playing outside, it does not accurately reflect the logical relationship presented in the initial condition.

D) If the children are not playing outside then the television is not off

This statement incorrectly suggests that the television being off is a necessary condition for the children not playing outside. It does not follow from the original statement and misinterprets the logical relationship.

Conclusion

Option B is definitively correct because it accurately represents the contrapositive of the original statement, preserving the logical relationship. All other options either misinterpret the original statement or fail to establish a valid logical equivalence, reinforcing why B is the only correct choice.