Question

If A and B are subsets of a set X, prove that A\B = A intersects complement of B.

Answer 1

Ok this is my guess ... feel free to critique

Suppose \(A\subseteq X\) and \(B \subseteq X\) such that every element of A and B are elements of X

Suppose \(y \in A\) and \(y \notin B\) so\( y \in A−B\)

Hence \(y \in A\) and \(y \in B^c\) so \(y \in A \cap B^c\)

Therefore \(A−B=A \cap B^c\)

Answer 2

Im just wondering if Im missing a line in there
Like maybe im somehow supposed to say that B^c is also a subset of X but im not sure that every element of B^c is an element of X

Answer 3

usually to prove an equality
you prove one is a subset of the other by supposing there is an element in A-B and then showing it is also in A and not B.
Then you suppose there is an element in A and not B and show it is also in A-B.

Answer 4

ohhhh yeaaaaaaa

Answer 5

Like I would rewrite what you have so far as this:
We want to prove
\[A-B \underline{\subset }A \cap B^c \\ \text{ Let } y=A-B \\ \text{ then by definition of } A-B \\ y \in A \text{ and } y \not \in B \\ \text{ so that implies } \\ y \in A \text{ and } y \in B^c \\ \text{ therefore } y \in A \cap B^c\]

now you show the other direction
show
\[A \cap B^c \underline{\subset}A-B\]
first line:
\[\text{ Suppose } x \in A \cap B^c \]

--
though I guess you could say some of the things we are saying seem redundant :p

Answer 6

you can also prove this using a truth table

Answer 7

ya proving sometimes tend to be redundant but i prefer more steps then less cuz its easier to follow Otherwise im like .... waitttt how did they get there???
Anyways thanks :)

Answer 8

and you still have to go that other direction

Answer 9

Ya the other direction is pretty much similar

Answer 10

oh wait you know what?
I was going to prove this by truth table
and I just remember that I have always defined A-B as A and not B
:p
so truth table with not work since I used that as definition

Answer 11

it is actually kinda weird to prove it non truth table wise
because we are still saying A-B means A and not B

Answer 12

ik but i remember when i used to prove this crap we wld basically just do that
ANDDD
A-B mean the element is in A and Not In B

Answer 13

oops I didn't realize this wasn't your question

Answer 14

lol yea just reviewing this stuff cuz im bored -.-
Or rather this a form of procrastination
I prefer studying leisurely than studying stuff I must know
Got a huge exam and this is a way of just forgetting abt it ;)
Kinda sad lol

Answer 15

just change your statements to iff and you are done ;)

\(x\in A-B \iff x\in A \text{ and } x\notin B \iff x\in A \text{ and } x\in B^C\iff x\in A\cap B^c\)