# Math Help - (set theory) I don't understand the answer to this past exam question.

1. ## (set theory) I don't understand the answer to this past exam question.

The first two images are the question

The last one is the explanation from the prof which I do not understand.

I don't understand why the answer shouldnt be D (it's E)

To me, the set means "All elements that fall outside of A, plus all elements falling outside of R/V/R&V." Which would make the answer to the problem choice D Can someone translate into english what the set means, and possibly explain in another way why the answer is E?

2. Originally Posted by smallfry
The first two images are the question

The last one is the explanation from the prof which I do not understand.

I don't understand why the answer shouldnt be D (it's E)

To me, the set means "All elements that fall outside of A, plus all elements falling outside of R/V/R&V." Which would make the answer to the problem choice D Can someone translate into english what the set means, and possibly explain in another way why the answer is E?

$A^c=\text{all undergrad students not in the faculty of arts}$

$(R\cap V)^c\equiv R^c\cup V^c \ \text{by DeMorgan's law}$

$R^c=\text{all undergrads who aren't living in residence}$

$V^c=\text{all undergrads who don't own their own vehicals}$

Unioning $R^c, \ V^c$ we have all undergrads who aren't living in residence or don't own their own cars.

And finally, all undergrads who aren't living in residence or don't own their own cars or aren't an undergrad in the faculty of arts.

Make sense?

3. I did not look at the explanation.
But here is the reason it is E.
$A^c \cup \left( {R \cap V} \right)^c \equiv A^c \cup R^c \cup V^c \equiv \left( {A \cap R \cap V} \right)^c$