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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

A = {1, 2, 3, 4}

$space space space space space space space space space space space space space space space space space space space space space space straight a element of straight A comma space straight b element of straight A space rightwards double arrow space left parenthesis straight a comma space straight b right parenthesis space element of straight A space straight x space straight A$

Now, A x A = {1, 2, 3, 4} x {1, 2, 3, 4}

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

(i) S = $space space space left curly bracket left parenthesis straight a comma space straight b right parenthesis space colon space straight a space element of straight A comma space straight b element of straight A comma space$ a divides b)

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

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

$rightwards double arrow$ B = {2, 3}

(iii) S x B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4) X (2, 3)

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

(4, 4, 2), (4, 4, 3)}

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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 :

<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>

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

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