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

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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 $space space space space space space space space space space space space straight A cross times straight A equals left parenthesis straight A cross times straight B right parenthesis intersection left parenthesis straight B cross times straight A right parenthesis$

Here, A = {2, 3}, B = {1, 2, 3, 4}

$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ A x A = {2, 3} x {1, 2, 3, 4} ...(i)

= {(2, 1), (2, 2), (2, 3), (2,4), (3, 1),(3, 2), (3,3),(3,4)} ...(ii)

Also, B x A = {1, 2, 3, 4} x {2, 3} = {(1, 2), (1, 3), (2, 2), (2, 3), (3, 2), (3, 3), (4, 2), (4, 3)} ...(iii)

From (ii) and (iii), we have

$space space space space space space space space space space space space open parentheses straight A space straight x space straight B close parentheses intersection left parenthesis straight B space straight x space straight A right parenthesis space equals space left curly bracket left parenthesis 2 comma space 2 right parenthesis comma space left parenthesis 2 comma space 3 right parenthesis comma space left parenthesis 3 comma space 2 right parenthesis comma space left parenthesis 3 comma space 3 right parenthesis right curly bracket space$ ....(iv)

Hence, from (i) and (iv),

$space space space space space space space space space space space space space space straight A space straight x space straight A space equals space left parenthesis straight A space straight x space straight B right parenthesis intersection left parenthesis straight B space straight x space straight A right parenthesis$

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25.

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

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