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

108 Views

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

88 Views

23.

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

109 Views

24.

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

101 Views

25.

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

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

148 Views

26. Let A, B, C and D be any non-empty sets. Prove that

$space space space space space space space space space space space space space open parentheses straight A space straight x space straight B close parentheses intersection left parenthesis straight C space straight x space straight D right parenthesis space equals space left parenthesis straight A intersection straight C right parenthesis space straight x space left parenthesis straight B intersection straight D right parenthesis$

<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>

rightwards double arrow

space space space space space space space space space space space space space space space space space left parenthesis straight x comma straight y right parenthesis element of open parentheses straight A space straight x space straight B close parentheses space and space left parenthesis straight x comma space straight y right parenthesis space element of open parentheses straight C cross times straight D close parentheses

rightwards double arrow space

space space space space space space space space space space space space open parentheses straight x element of straight A space and space straight y element of straight B close parentheses space and space open parentheses straight x element of straight C space and space straight y element of straight D close parentheses

rightwards double arrow

space space space space space space space space space space open parentheses straight x element of straight A space and space straight x element of straight C close parentheses space and space left parenthesis straight y element of straight B space and space straight y element of straight D right parenthesis

rightwards double arrow

<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>

∴ $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis straight x comma space straight y right parenthesis space element of space left parenthesis straight A space straight x space straight B right parenthesis intersection left parenthesis straight C space straight x space straight D right parenthesis rightwards double arrow left parenthesis straight x comma straight y right parenthesis element of left parenthesis straight A intersection straight C right parenthesis straight x left parenthesis straight B intersection straight D right parenthesis$

$rightwards double arrow$ $space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis straight A space straight x space straight B right parenthesis intersection left parenthesis straight C space straight x space straight D right parenthesis space bottom enclose subset of space left parenthesis straight A intersection straight C right parenthesis space straight x space left parenthesis straight B intersection straight D right parenthesis$ ....(i)

Now, $space space space space space space space space space space space space space space space space let space left parenthesis straight x comma straight y right parenthesis space element of left parenthesis straight A intersection straight C right parenthesis space straight x left parenthesis straight B intersection straight D right parenthesis space be space any space element.$

$rightwards double arrow$ $space space space space space space space space space space space space straight x element of straight A intersection straight C space and space straight y element of straight B intersection straight D$

$rightwards double arrow space$ $space space space space space space space space space space space left parenthesis straight x element of straight A space and space straight x element of straight C right parenthesis space and space left parenthesis straight y element of straight B space and space straight y element of straight D right parenthesis$

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

$rightwards double arrow$ $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses straight x comma straight y close parentheses element of straight A cross times straight B space and space open parentheses straight x comma straight y close parentheses element of C space x space D$

$rightwards double arrow$ $space space space space space space space space space space space space space space space space space space open parentheses straight x comma straight y close parentheses element of left parenthesis straight A cross times straight B right parenthesis intersection left parenthesis straight C cross times straight D right parenthesis$

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

Hence, from (i) and (ii)

$space space space space space space space space space space space space space space space space space space space space space space space space space open parentheses straight A space straight x space straight B close parentheses intersection left parenthesis straight C space straight x space straight D right parenthesis space equals space left parenthesis straight A intersection straight C right parenthesis space straight x space left parenthesis straight B intersection straight D right parenthesis$