[MATH SETS] can you conclude that A=B if A, B, and C are sets such that....?

Question

such that...

a. A U C = B U C
b. A n C = B n C
c. A U C = B U C and A n C = B n C

im having trouble understanding how to prove that these are equal. thanks in advance for whoever helps me. :)

Answer 1

Let A = {1}, B = {2}, and C = {1,2}. Then AUC = C = BUC, but A and B are not equal. So, a. is false.



Let A = {1,2}, B = {1,3}, and C = {1}. Then, AnC = C = BnC, but A and B are still not equal. So, b. is false.



Suppose A U C = B U C and A n C = B n C. Let x ∊ A. Then x ∊ A U C, so by assumption, x ∊ B U C. Thus, x ∊ B or x ∊ C. If x ∊ B, then we are done. If x ∊ C, then we have x ∊ A n C. By assumption, this means x ∊ B n C, which means x ∊ B. So, in every case, A ⊆ B. A similar argument will show that B ⊆ A, so A = B.

Answer 2

I would say C, because it makes the most sense. I can't prove it, but here's how I figured:



If A and B both equal the same thing, then if you replace A with B (such as seen in both A U C = B U C and A n C = B n C) then it would equal the exact same thing. If you notice, besides A and B, the rest of the set is exactly the same. So it makes C the most logical answer. I hope this helps!

Answer 3

c is right,

you can do it like this

IF A= (12345) B=(123) C=(45) AndA U C = B U C

But A =/B So " a" is wrong

IF A=(1234) B=(234) C=(4) and A n C = B n C

But A =/B So " b" is wrong

":a "and "b "is wrong .there is only one right answer in the question,

so choose C.



I donot konw how to prove A U C = B U C and A n C = B n C .conclude that A=B

You can try this way to understand this qusetion, why itis right you can ask your theater for help.

Answer 4

First we ought to locate the identity element, e, of team G. a * e = a a + e - ae = a e - ae = 0 e(a million - a) = 0 e = 0 or a million - a = 0 e = 0 or a million = a we won't be able to have a = a million for the reason that a is in G and G = Q - {a million}. Then e = 0. enable a be a factor of G. Then a^(-a million) a * a^(-a million) = e a + a^(-a million) - aa^(-a million) = 0 a^(-a million) - aa^(-a million) = -a a^(-a million) [a million - a] = -a a^(-a million) = -a / [a million - a] = -a / -[a - a million] = a/(a-a million)

Answer 5

well, if AUC = BUC, then A-C = B-C.



(suppose x is in A-C.x is in A, so x is in AUC. since AUC = BUC, x is in BUC. so x is either in B or C.

since x is not in C, x must therefore be in B, and so is in B-C. the converse is similar).



now, if in addition A∩C = B∩C, then A = (A-C)U(A∩C) = (B-C)U(B∩C) = B.