For any set A,B,C..... (elementary algebra)

Question

For any set A,B,C in a universal set U, (A/B)C = A/(BC).....State true or false and give a proof supporting your answer.

Answer 1

False, because if, for example, A=2, B=3, C=4, then A/B=2/3 and (A/B)C=(2/3)*4=8/3; but BC=3*4=12 and A/(BC)=2/12=1/6. The numbers 2, 3, 4, in N, are part of U, because N is a subset of U so (A/B)C = A/(BC) cannot be universally true.

Comments

But according to question.... A B & C are subsets of U.... So does your explanation implies...??

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Sorry, I assumed they were elements rather than sets. However, a set can consist of one element or, as I thought, A, B and C were individual elements of the U set, forming a set { A B C ... } rather than sets of elements within U. The question then arises; if these are sets what is the meaning of division or multiplication in the context? How do you multiply and divide sets of more than one element? The key thing, I think, is that C is moved from numerator to denominator position, so whatever the meaning, C is in a different arithmetic position, which would make the statement false in any case. If you take my answer to be a special case of a 1-element set, the statement is false, and to be true it must apply to all sets, including 1-element sets. I hope I haven't confused you!

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Well you did a little but I'm going to take my chances.... Thank you for your help and it would be great if you could explain about it more but it's fine as well if you don't want to coz I have a book for that.... Hence, I'm going to assume these sets are singleton and apply your main answer....

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Sorry for any confusion and for the delay in replying (it took me a while to find your comment!). What struck me about your question was division and multiplication: (A/B)C and A/(BC) I took to be the normal arithmetic application of division and multiplication, which applies to numbers rather than sets (at least, in my limited experience with sets); that's why I assumed A, B and C were just numbers, single elements within U. So my line of thought was that (A/B)C expands to AC/B, with C in the numerator formed by the product of A and C; while A/(BC) contains the product of B and C in the denominator, so the two expressions are different.

Since it's over 50 years since I studied mathematics you are probably more familiar with sets than I am, and perhaps it's possible to apply multiplication and division to whole sets (as in matrices, for example); but I personally don't know how it's done.

I would be interested to know if you had a different interpretation. Yes, I will willingly offer explanations you need, as long as I understand the subject. You only have to ask.

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Thank you very much on your support.... And I'll inform you as soon as I find an alternative answer or the same