Comparing T/F for Three Sets: A, B & C

[EvLer]

Here is this problem I actually don't know how to approach:

which of the following are T/F for all sets A, B, C
a. A U (B x C) = (A U B) x (A U C)
i think this one is false, i worked it out on a small example
b. A x (B intersection C) = (A x B) intersection (A x C)
c. A x (B x C) = (A x B) x C

where the "x" is cartesian product.
Any idea how to approach this systematically or just a helpful explanation is very much appreciated.

 

[d_leet]

The typical method for proving set equalities is to choose some arbitrary element of one side of the equality call it x then show that because of it belonging to one side, and the properties of unions, intersections, and cartesian products that it belongs to the other side of the equality. This shows that the first side is a subset of the second, so to prove them equal you need to show that the second side is also a subset of the first by using the same method.

I'll give you an example.
Let's say I want to prove that

$$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$
Then I want to start with
$$let \ x \ \epsilon \ A \cap (B \cup C)$$
then
$$x \ \epsilon \ A \ and \ x \ \epsilon B \cup C$$
from that I know that
$$x \ \epsilon A \ and \ ( x \ \epsilon B \ OR \ x \ \epsilon C )$$
Which means that
$$x \ \epsilon \ A \cap B \ OR \ x \ \epsilon \ A \cap C$$
And so
$$x \ \epsilon \ (A \cap B) \cup (A \cap C)$$

this proves that $$A \cap (B \cup C) \subset (A \cap B) \cup (A \cap C)$$

And to finish the proof I would just need to start with some other arbitrary element, say y, is in ( A intersect B) U (A intersect C) and perform the same procedure to show that this is a subset of A intersect ( B U C) then since each side is a subset of the other they must be equal, and the proof would be complete.

Note that when doing a proof like this witha cartesian prodcut to remeber that the elements of a cartesian product are pairs of elements (x, y) so that if this was a member of the cartesian product X x Y then x is an element of X and y is an element of Y.