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January 19

A subset B iff A intersection of B = A

I don't understand how can I prove this problem. I need book or name of book can help me. Because I don't understand how to prove "iff" case. — Preceding unsigned comment added by 151.236.179.245 (talk) 19:10, 19 January 2018 (UTC)[reply]

Please clarify I don't understand how to prove "iff" case. A⊆B ⇔ A∩B=A means A⊆B ⇒ A∩B=A and A∩B=A ⇒ A⊆B. (See If and only if.) Which inference were you able to prove and which inference eludes you? -- ToE 19:51, 19 January 2018 (UTC)[reply]

For resources, our Wikibooks project has a text on Abstract Algebra with an introductory chapter on sets, wikibooks:Abstract Algebra/Sets. -- ToE 19:58, 19 January 2018 (UTC)[reply]

I need a name of book in sets.please... — Preceding unsigned comment added by 151.236.179.205 (talk) 10:18, 20 January 2018 (UTC)[reply]

For Wikipedia articles on the subject matter, see Category:Set theory and Category:Basic concepts in set theory.

For Wikipedia articles on books on the subject, see Special:Search/incategory: "Mathematics textbooks" incategory: "Set theory".

For a list of textbooks on set theory compiled by the participants of the mathematics stack exchange, see Reference request: Textbooks on set theory.

For assistance with the problem at hand, I would recommend the introductory chapter (or chapter 0) of an introductory text to abstract algebra.

You may benefit from a text teaching basic concepts and strategies for mathematical proofs. This Quora discussion mentions several options, some of which are available for free online. One you may wish to check out is Mathematical Reasoning: Writing and Proof by Ted Sundstrom of Grand Valley State University, hosted by GVSU here. Chapter 5 is dedicated to set theory and looks to be at an appropriate level. -- ToE 11:24, 20 January 2018 (UTC) Ted Sundstrom's Mathematical Reasoning is on the list of approved textbooks from the American Institute of Mathematics Open Textbook Initiative. The complete list -- currently 47 texts in 18 categories -- is here. -- ToE 20:31, 20 January 2018 (UTC)[reply]

To prove an iff statement, you need to do two things: first, show that A⊆B ⇒ A∩B=A, and second that A∩B=A ⇒ A⊆B. You'll probably want to start by assuming that A⊆B, and therefore any element a $\in$ A is also contained in B. From there, try to deduce what that means for the intersection of A and B. That will take care of the first step, and the proof of the second step will be similar.OldTimeNESter (talk) 16:43, 20 January 2018 (UTC)[reply]

In fact you don't have anything to prove, you just need to understand what ⊆ ∩ and = mean in the context of sets pm a 23:35, 22 January 2018 (UTC)[reply]

While the proof involves only a simple manipulation of the formal definitions of subset, intersection, and equality, the specified biconditional statement is not axiomatic, so it is perfectly reasonable for an elementary textbook to request such a proof. A student having difficulty with such a simple proof might not understand the concepts involved, but they are just as likely to either not be familiar the formal definitions of the simple concepts or not understand the mechanics of constructing a proof. Hence my recommendation of Sundstrom's text which contains some elementary set theory but is primarily focused on the mechanics of proof writing. -- ToE 18:27, 24 January 2018 (UTC)[reply]

There is also yet another equivalent statement using the union: A⊆B if and only if A∪B=B. Putting everything together, A⊆B iff A∩B=A iff A∪B=B. One more equivalent statement is A\B=∅, where the backward slash denotes the set difference. GeoffreyT2000 (talk) 02:30, 25 January 2018 (UTC)[reply]