( ' × ' = Cartesian product ' ∩ ' =intersection }
How can these be proved
a) ( A - B ) × C = ( A × C ) - ( B × C )
b) A ∩ B = B - ( B - A )
(they are correct, right ?)
I didn't find any example that would prove them wrong.
What laws can I use to prove them right ?
Would be helpful if you could give me the steps needed.
Thanks.
2007-09-27 08:00:38 · 4 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
a) Suppose the ordered pair (x,y) belongs to (A - B ) × C. Then, x belongs to A - B and y belongs to C. Since x belongs to A - B, x belongs to A and does not belong to B. This implies that (1) (x,y) belongs to A x C and (2) (x,y) does not belong to B x C (or x would belong to B). So, x belongs to (A x C) - (B x C), which implies (A - B ) × C is subset of (A x C) - (B x C).
Now suppose, on the other hand, that (x,y) belongs to (A × C ) - ( B × C ). Then, (x,y) belongs to A x C and does not belong to B x C. This implies x belongs to A and y belongs to C. Since y belongs to C but (x,y) is not in B x C, it follows x is not in B. Therefore, we conclude x is in A - B and y is in C, that is, (x,y) is in (A - B) x C. Hence, ( A × C ) - ( B × C )
is subset of (A - B) x C
We conclude (A - B ) × C = ( A × C ) - ( B × C )
b) Observe that B - ( B - A) = B ∩ (B - A)', where ' means the complement. So, B - ( B - A) = B ∩ (B ∩ A')'. By de Morgan's laws, it follows that B - ( B - A) = B ∩ (B' U A) = (B ∩ B') U (B ∩ A) = {} U (B ∩ A) = B ∩ A = A ∩ B, proving the proposition.
2007-09-27 08:34:58 · answer #1 · answered by Steiner 7 · 0⤊ 0⤋
In general, showing 2 sets are equal is done in 2 parts. Show that one is contained in the other, and then vice versa. So your approach should be to show that every element of the left side is included in the right side, and every element of the right side is included in the left side.
2007-09-27 15:07:53 · answer #2 · answered by Larry B 2 · 0⤊ 0⤋
analyse two elements (calle them x and y) where one element belongs to the set on the left of the = and the other does not.
e.g., x belongs to (A-B)xC
what does that tell you about x?
With that knowledge, can you tell if x belongs to the set on the right of =.
then let y "not belong" to (A-B)xC and show that this makes it impossible for y to belong to (AxC)-(BxC).
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In summary:
if x belongs to A â© B, then x MUST belong to B - ( B - A )
If you find one single x that does not fit this, then you will have disproved the equality.
2007-09-27 15:10:53 · answer #3 · answered by Raymond 7 · 0⤊ 0⤋
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2007-09-27 15:04:05 · answer #4 · answered by SAK S 1 · 0⤊ 0⤋