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Relations and Functions

Short Answer Type

11. Let A = {–1, 1}, B = {2, 3}, C = {3, 5} be subsets of real numbers.

n(A X A X A)

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12. Let A = {1,2, 3, 4}

space space space space space space space space space space space straight S space equals space left curly bracket left parenthesis straight a comma space straight b right parenthesis space colon space straight a element of straight A comma space straight b space element of space straight A comma space straight a space divides space straight b right curly bracket

space space space space space space space space straight B space equals space left curly bracket straight x space colon space straight x element of straight A comma space straight x space is space straight a space prime right curly bracket

Write (i) S explicity, (ii) B explicity, (iii) S x B

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13.

If A × B = {(a, x),(a , y), (b, x), (b, y)}. Find A and B.

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14.

The Cartesian product A × A has 9 elements among which are found (–1, 0) and (0,1). Find the set A and the remaining elements of A × A.

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15. Let A = {I, 2, 3}, B = {2, 3, 4} and C = {4, 5}. Verify that :

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<pre>uncaught exception: <b>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</b><br /><br />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<br />#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>

space space space space space space space space space space space space straight A cross times open parentheses straight B intersection straight C close parentheses space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket space straight X space left curly bracket 4 right curly bracket space equals space left curly bracket left parenthesis 1 comma space 4 right parenthesis comma space left parenthesis 2 comma space 4 right parenthesis comma space left parenthesis 3 comma space 4 right parenthesis right curly bracket

...(i)

A x B = {1, 2, 3} X {2, 3, 4}

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

A x C = {1, 2, 3} x {4, 5}= {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} ....(iii)

From (ii) and (iii), we have

space space space space space space space space space space space space space space space space left parenthesis straight A space straight X space straight B right parenthesis intersection left parenthesis straight A cross times straight C right parenthesis space equals left curly bracket left parenthesis 1 comma space 4 right parenthesis comma space left parenthesis 2 comma space 4 right parenthesis comma space left parenthesis 3 comma space 4 right parenthesis right curly bracket

Hence, from (i) and (iv)

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16. Let A = {I, 2, 3}, B = {2, 3, 4} and C = {4, 5}. Verify that :

$space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight A cross times open parentheses straight B union straight C close parentheses equals left parenthesis straight A cross times straight B right parenthesis union left parenthesis straight A cross times straight C right parenthesis$