•Are A∪B=A∩B and A=B completely equivalent? •Simplify A∪B=A∩B to A=B using set operations and their properties.


Question and Answers Forum

All Questions Topic List
Set Theory Questions
Previous in All Question Next in All Question
Previous in Set Theory Next in Set Theory
Question Number 1698 by Rasheed Ahmad last updated on 01/Sep/15

$∙ A r e A \cup B = A \cap B a n d A = B$ $c o m p l e t e l y e q u i v a l e n t ?$ $∙ S i m p l i f y A \cup B = A \cap B t o A = B$ $u s i n g s e t o p e r a t i o n s a n d t h e i r$ $p r o p e r t i e s .$
	Answered by 123456 last updated on 01/Sep/15

	$if A \cup B = A \cap B them A = B$ $we have that A \cup B = A \cap B them \forall x, x \in A \cup B, x \in A \cap B$ $them supose that A \neq B, without loss$ $of generality suppose tbat ∣ A ∣>∣ B ∣$ $x \in A, x \notin B$ $x \in A \cup B, x \notin A \cap B (contradition)$ $hence A = B$ $and if A = B, the$ $A \cup B = A \cup A = A$ $A \cap B = A \cap A = A$ $A \cup B = A \cap B$ $so$ $A \cup B = A \cap B \Leftrightarrow A = B$
	Answered by 123456 last updated on 01/Sep/15

	$A \cup B = A \cap B$ $(A \cup B) \cup A = (A \cap B) \cup A$ $A \cup B = A$ $(A \cup B) \cup B = (A \cap B) \cup B$ $A \cup B = B$ $A = B$
	Commented by 123456 last updated on 01/Sep/15

	$(A \cup B) \cap A = (A \cap B) \cap A$ $A = A \cap B$ $(A \cup B) \cap B = (A \cap B) \cap B$ $B = A \cap B$
	Commented by Rasheed Ahmad last updated on 01/Sep/15

	$G^{o o} {DD}_{eductio} N_{!}^{!}$
	Commented by 123456 last updated on 02/Sep/15
	$also we can extend it to ∪_(i=1) ^n A_i =∩_(i=1) ^n A_i ⇔A_i =A_j ,i∈{1,...,n},j∈{1,...,n} n∈N^∗$
	$also we can extend it to$ $\underset{i = 1}{\overset{n}{\cup}} A_{i} = \underset{i = 1}{\overset{n}{\cap}} A_{i} \Leftrightarrow A_{i} = A_{j}, i \in {1, . . ., n}, j \in {1, . . ., n}$ $n \in N^{*}$
	Commented by Rasheed Soomro last updated on 03/Sep/15

	$G e n e r a l i z a t i o n! V . G o o d!$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum