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
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.
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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