Unsupported. If Jamil’s cat is outside, then one of his family members is outside. And if one of his family members is outside, then it’s sunny or hot. Jamil’s family member and his cat could be outside when it’s below freezing but still sunny.

Unsupported. If Jamil’s cat is outside, then it’s sunny or hot outside. But the fact that his cat is not outside tells us nothing about whether it’s sunny or hot. It’s possible that it is sunny or hot outside, but Jamil just didn’t want to let his cat out.

Unsupported. “Sunny or hot” is not a sufficient condition for anything. We know that Jamil’s family members (and thus his cat) will only be outside when it’s sunny or hot. But the fact that it’s sunny or hot tells us nothing about whether his family members or his cat are out.

Unsupported. If at least one family member is outside, then it’s sunny or hot outside. This doesn’t necessarily mean that the cat is also outside. We just know that if the cat is out, then at least one family member is also out. (D) gets this backward.

Very strongly supported. By taking the contrapositive of the conditional chain below, we see that if it’s not sunny or hot, then Jamil’s family members aren’t outside. And if they’re not outside, then Jamil won’t let his cat out. So if it’s not sunny or hot, his cat can’t be out.