Math, asked by master9613, 10 months ago

if Ax Bis a equal subset ofCxD and A*B is not equal to null set then-
prove that A is equal subset of C and B is also a equal subset of D.​

Answers

Answered by PantherSpider
6

Answer:

Step-by-step explanation:

First we consider the case:(A×B)⊂(C×D)Let a and b be two elements, such thata∈A  and b∈BThen, by definition of cartesian product of sets (a,b)∈A×B⇒(a,b)∈C×D              [As, (A×B)⊂(C×D)]⇒a∈C  and  b∈DThus, we haveIf   a∈A⇒a∈C, therefore  A⊂C.If  b∈B⇒b∈D, therefore  B⊂D.Hence, if A×B⊂C×D, then A⊂C  and  B⊂D.Now, consider the case when (A×B)=(C×D) , then (A×B)⊂(C×D)   and    (C×D)⊂(A×B)we have already proved thatif , (A×B)⊂(C×D), then  A⊂C  and  B⊂D.         ...........(i)Now when  (C×D)⊂(A×B), then  let c  and  d be two elements such thatc∈C  and   d∈D⇒(c,d)∈C×D⇒(c,d)∈A×B                                    [As, (C×D)⊂(A×B)]⇒c∈A   and  d∈BTherefore,C⊂A  and  D⊂B                                          ..........(ii)From (i) and (ii), we haveA⊂C  and  C⊂A⇒A=CB⊂D  and  D⊂B⇒B=DHence, if  (A×B)=(C×D)  then A=C  and  B=D.

Answered by om624870
0

Answer:

here, let A € {1,2} and B€ {3,4} & let C€{1,2,5} and D€{3,4,6}

Then,

A×B = (1,3),(1,4),(2,3),(2,4)

C×D =(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(5,3),(5,4),(5,6)

Thus A×B is made up of with the elements of C×D.And,

A×B is the subset of C×D (given)

we also conform that values we have taken is correct.

Now we can clearly see that A is made up of with the elements of C

Then A is the subset of C

and, B is made of with the elements of D thus, B is the subset of D (proved)

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