Relations and Functions

Relations and Functions

Question

Taking the set of natural numbers as the universal set, write down the complements of the following sets :

(i) {x : x is an even natural number}           (ii) {x : x is an odd natural number}

(iii) {x : x is a positive multiple of 3}            (iv) {x : x is a prime number}

(v) {x : x is a natural number divisible by 3 and 5}  

(vi) {x : x is a perfect square}                  (vii) {x : x is a perfect cube}

(viii) { x : x + 5 = 8 }                               (ix) { x : 2x + 5 = 9}

(x) { x : x ≥ 7 }                                       (xi) {x : x ∈ N and 2x + 1 > 10}

Answer
space space space space space space space space space space space space space space space space space space space straight U space equals space left curly bracket straight x space colon space straight x element of straight N right curly bracket space equals space left curly bracket 1 comma space 2 comma space 3 comma space 4 comma space 5 comma space 6 comma space......... right curly bracket
(i) A = {x : x is  na even atural number} = <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>

∴ A ' = U - A = <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>
              
                 = {x : x  is an odd natural number}

(ii)  A = {x : x is an odd number}
A' = U - A = space space left curly bracket space straight x thin space colon space straight x space element of space straight N right curly bracket space - {x : x is an odd number}
= {x : x is an even natural number} = <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>

(iii) A = {x : x is a positive multiple of 3}
A' = U - A = {1, 2, 3, 4, 5, 6, .......} - {3, 6, 9, 12, .........}
= {1, 2, 4, 5, ...........} = space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N, x is not a multiple of 3}

(iv)    A = {x : x is a prime number} = space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N, x is priime number}
A' = U - A = space space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N space right curly bracket - space space space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N comma x is a prime number
space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N, x is not a prime number}

(v)    A = space space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N, x is divisible by 3 and 5}
=space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N, x is divisible by 15.
A' = U - A  = space space space space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N right curly bracket space minus space space space left curly bracket space straight x thin space colon space straight x space element of space straight N, x is divisible by 15}
<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>, x is not divisible by 15}
=    space space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N, x is not divisible by 3 and 5}

(vi) A = { x : x is a perfect square}
A' = U - A = <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> - (x : x is a perfect square)
space space space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N, x is not a perfect square}

(vii)   A = {x : x is a perfect cube}
A' = U - A = <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> - {x : x is a perfect cube}
space space space space space left curly bracket space straight x thin space colon space straight x space element of space straight N, x is not a perfect cube}

(viii)  A = {x : x +5 = 8} = {x : x = 3} = {3}

∴   A' = U - A = {1, 2, 3, 4, 5, 6, .........} - {3} = {1, 2, 3, 4, 5, 6, .........}

∴   A' = space space space space left curly bracket space straight x space colon space straight x space element of space straight N comma space space straight x not equal to 3 right curly bracket

(ix)    A = { x : 2x + 5 = 9} = {x : 2x = 4} = {x : x = 2} = {2}
∴   A' =  U - A = {1, 2, 3, 4, 5, 6, ..........} - {3} = {1, 2, 4, 5, 6, ............}
space space space space space space space equals space left curly bracket straight x space colon space straight x space element of space straight N comma space straight x not equal to 2 right curly bracket

(x)     A = space space space space left curly bracket straight x space colon space straight x space greater or equal than space 7 right curly bracket
A' = U - A = space space space space space space equals space left curly bracket straight x space colon space straight x space element of space straight N right curly bracket space minus space left curly bracket space straight x space colon space straight x space greater or equal than 7 right curly bracket space equals space left curly bracket straight x space colon space straight x element of space straight N comma space straight x space less than 7 right curly bracket
= {1, 2, 3, 4, 5, 6}

(xi)       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 space space space space straight A space space space equals space left curly bracket straight x space colon space straight x space element of space straight N comma space 2 straight x space plus space 1 space greater than space 10 right curly bracket space equals space left curly bracket straight x space colon space straight x element of straight N comma space 2 straight x space greater than space 9 right curly bracket

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 equals comma space left curly bracket straight x space colon space straight x element of straight N comma space straight x space greater than space 4 1 half right curly bracket space equals space left curly bracket 5 comma space 6 comma space 7 comma space 8 comma space...... right curly bracket

∴    A' = U - A = {1, 2, 3, 4, 5, 6, 7, 8, .........} - {5, 6, 7, 8, ..........}
      = {1, 2, 3, 4}

 

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