Question

prove (A\B) ^ (C\D)= (A^C)\ (B^D)

Answer 1

\[(\frac{A}{B})^\frac{C}{D}=\frac{A^\frac{C}{D}}{B^\frac{C}{D}}\]

Answer 2

oh completely sorry...this is for real analysis

Answer 3

can you get me more details on this notation you used

Answer 4

what is A,B,C and D?

Answer 5

the \ is "compliment" and the ^ is "intersection"

Answer 6

oh

Answer 7

and A,B,C,D are subsets of universal set U

Answer 8

compliment: my understanding is that means elements not in
do you mean A-B

Answer 9

yeah that is the general notation, for some reason my book uses \

Answer 10

ok so i'm going try to prove
\[(A-B) \cap (C-D)=(A \cap C)-(B \cap D)\]

Answer 11

correct

Answer 12

ok let me look at it for a few

Answer 13

well first of all you know we have two ways to go to prove the equality
we have to prove each is a subset of the other

Answer 14

if you have any advice for me on this course too i would appreciate it, im struggiling

Answer 15

what text are you using?

Answer 16

Analysis with an introduction to proof by Steven R. Lay

Answer 17

hey satellite tell me if i'm thinking right
i'm a bit tired...

Answer 18

if x is an element of A-B, then x is an element of A but not B
right?

Answer 19

i think this sounds right

Answer 20

right

Answer 21

ok i have the first part one sec

Answer 22

you got it i think. do it element wise

Answer 23

suppose \[x \in (A-B) \cap (C-D) => x \in A-B \& x \in C-D\]

Answer 24

true

Answer 25

\[=> x \in A and x \not{\in} B \& x \in C and x \not{\in} D\]

Answer 26

my problem is this is the first time having 4 sets, can't draw a venn diagram and don't know how to do the proof

Answer 27

now we have to go the other way

Answer 28

satellite be my guest

Answer 29

but x is not an element of B

Answer 30

\[=>x \in (A \cap C)-(B \cap D)\]

Answer 31

oh ok

Answer 32

there happy lol

Answer 33

\[(A-B) \cap (C-D)=(A \cap C)-(B \cap D)\]
\[(A\cap C)-(A\cap D)-(B \cap C)-(B\cap D)=(A \cap C)-(B \cap D)\]
hmmmm

Answer 34

do the logic rules apply to sets?

Answer 35

yes they do

Answer 36

i have to say i am struggling with the other way.

Answer 37

i think this other way is a bit more trickier

Answer 38

lol

Answer 39

you have a few cases you have to consider going the other way

Answer 40

\[(A-B) \cap (C-D)=(A \cap C)-(B \cap D)\]

is that first one (A or notB) ?

is that first one (A not B) ?

is that first one (A and notB) ?

Answer 41

you get to a part in the proof where you have y is not an element of B intersect D
this could mean that y is not in both sets
this could mean y is in one set but not the other

Answer 42

these are the two cases that i see to consider

Answer 43

or is (A-B) mean (A n C n D)?

Answer 44

\[A-B=\{x \in A| x \not{\in} B\}\]

Answer 45

isn't
\[A-B=A\cap B^c\]?

Answer 46

yes the first one is A not B

Answer 47

you mean the difference of the sets A and B?

Answer 48

and satellite yes I think that is right, B superscript c is U/B

Answer 49

then i think that would make life easier

Answer 50

no just that (A-B)=A not B

Answer 51

lol .... just trying to understand what its saying to begin with seems to be a challange

Answer 52

\[A-B=\{x\in A| x\not{\in} B\}\]

Answer 53

copy cat

Answer 54

which to me means
\[\{x|x\in A, x\not{\in}B\}=A\cap B^c\]

Answer 55

anyone consider it to be false? can we give a counter example?

Answer 56

yes A intersect not B
but i can't figure out what A not B means

Answer 57

well mathty there has to be a counterexample since going the other way y may not be in B but in D or the other way around
so i bet i can come up with one

Answer 58

So, x is an element of A and notB; and element of C and notD is equivalent to saying: x is an element of A and D, but not an element of B and C

Answer 59

A not B means all the elements that are in A and not B, if you draw a venn diagram you will see that a and b overlap

Answer 60

ok i know what A and not B means
i just didn't know what A not B means

Answer 61

of course, how you draw a venn diagram for 4 sets, Im not sure

Answer 62

i am thinking maybe the reason it is hard is that it is not right,
x is an elem