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Relations and Functions

Short Answer Type

21.
Let A = {a, b, c}, B = {d, e, f}, C = {g, h, k}, L = {b, c}, M= {d, e} and N = {g, k}.

verify that $space space space space space space space space open parentheses straight L cross times straight M cross times straight N close parentheses subset of open parentheses straight A cross times straight B cross times straight C close parentheses$ .

A x B x C = {(a, d, g), (a, d, h), (a, d, k), (a, e, g), (a, e, h), (a, e, k), (a, f, g), (a, f, h), (a, f, k), (b, d, g), (b, f, h), (b, f, k), (c, d, g), (c, d, h), (c, d, k), (c, e, g), (c, e, h), (c, e, k), (c, f, g), (c, f, h), (c, f k) ...(i)

L x M x N = {(b, d),m (b, e), (c, d), (c, e)x (g, k)}

= {(b, d, g), (b, d, k), (b, e, g), (b, e, k), (c, d, g), (c, d , k), (c, e, g), c, e, ,k) ...(ii)

From (i) a nd (ii), we observe that every member of L X M x N is also a member of A X B X C

∴ $space space space space space space space space space space space space space space space straight L space straight x space straight M space straight X space straight N space bottom enclose subset of space straight A space straight x space straight B space straight x space straight C$

Also, n(L x M x N) < n(A x B x C)

Hence, $space space space space space space space space space space space space space space space space space left parenthesis straight L space straight x space straight M space straight x space straight N right parenthesis space subset of space left parenthesis straight A space straight x space straight B space straight x space straight C right parenthesis$

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22. Let A = {a, b), B = {c, d}. Write A x B. How many subsets will A x B have? List them.

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23.

$space space space space space space space space space space space space space If space straight A bottom enclose subset of straight B comma space then space prove space that space straight A space straight x space straight A space equals space left parenthesis straight A space straight x space straight B right parenthesis intersection left parenthesis straight B space straight x space straight A right parenthesis$

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24.

Let A = {2, 3}, B = {1, 2, 3, 4}, verify that $space space space space space space space space space space space space straight A cross times straight A equals left parenthesis straight A cross times straight B right parenthesis intersection left parenthesis straight B cross times straight A right parenthesis$

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25.

If A and B are any two non-empty sets, then prove that

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