# Which one of the following, if substituted for the restriction that if music is taken, then neither physics nor theat...

on July 29 at 11:38PM

E is the contrapositive of the rule needed to substitute. Why wouldn't it work? The video didn't cover the rest of the choices. Thank you!!!

Shunhe on July 30 at 02:01AM

Hi @Partv,

Thanks for the question! So actually, (E) is not the contrapositive of the rule that’s being substituted! This one’s pretty tricky.

So the rule tells us “if music is taken, then neither physics nor theater can be taken.” Neither physics NOR theater means “not physics” AND “not theater.” So this would be diagrammed

M —> ~P & ~T

Now let’s take a look at (E). It tells us that if both physics and theater are taken, then music can’t be taken. This is

P & T —> ~M

Now, the question now becomes: when we negate ~P & ~T, do we get P & T? No, that’s not quite right. Remember that negating ~P & ~T means that we have to negate both the variables, and switch the “and” to an “or.” So the correct contrapositive of the original rule is

P v T —> ~M

In general, when we negate A & B, we negate both terms and switch the “and” to an “or” or vice versa. And if this is confusing, consider this example. Let’s say someone says “I have both apples and bananas,” which is A & B. How do we negate that statement? Well, it’s “I don’t have apples” or “I don’t have bananas.” If you have apples, but not bananas, you don’t have both apple and bananas. And same if you have bananas but not apples.

And so (E) isn’t actually the contrapositive, which explains why it doesn’t work.

Hope this helps! Feel free to ask any other questions that you might have.