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141.
Assume that P ( A ) = P ( B ). Show that A = B

Let $space space space space space space space space space space straight x space element of space straight A$

$rightwards double arrow$    $space space space space space space space space space space space space space space space space space space left curly bracket straight x right curly bracket space element of space straight P space left parenthesis straight A right parenthesis$

But, P(A) = P(B) (given) $rightwards double arrow$    $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left curly bracket straight x right curly bracket space space element of space straight P left parenthesis straight B right parenthesis$

$rightwards double arrow$ $space space space space space space space space straight x space element of space straight B$

So,    $space space space space space space space space space space space space space space space space space space space space space space space space straight x space element of space straight A rightwards double arrow space straight x element of space straight B$

∴ $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight A bottom enclose subset of space straight B$ ....(i)

Now,

space space space space space space space space space space space space space space space let comma space straight x space element of space straight B

<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>

But, P(B) = P(A) (given)

<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>

<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 rightwards double arrow space space space space space straight x space element of space straight A

So,

space space space space space space space space space space space space straight x element of straight B space rightwards double arrow space straight x space element of space straight A

∴ $space space space space space space space space space space space space space space space space space straight B space bottom enclose subset of space straight A$ ....(ii)

Hence, from (i) and (ii)

A = B

177 Views

142.

Is it true that for any sets A and B, P ( A ) ∪ P ( B ) = P ( A ∪ B )? Justify your answer.

124 Views

143.

Out of 500 car owners investigated, 400 owned car A, 200 owned car B, and 50 owned both cars A and B. Is the data correct?

155 Views

144. A survey was conducted on 800 students of a school with regards to their interest in playing cricket and hockey. 576 of the students said they liked to play circket where as 360 liked to play hockey. What is the least number of students who liked to play both the games?

121 Views

145.

There are 200 individuals with skin disorder. 120 of them are exposed to chemical C_1,50 to chemical C₂ and 30 to both C₁ and C₂. Find the number of individuals :

(i) exposed to chemical C₁, but not to chemical C₂.

(ii) exposed to chemical C₂ but not to chemical C₁

(iii) exposed to C, or chemical C₂ or both.

(iv) not exposed to any of the chemicals C₁ and C₂.

199 Views

146. X and Y are two sets such that X has 28 elements, Y has 32 elements and X ∩ Y has 10 elements. How many elements does X ∪ Y have?

136 Views

147. If A and B are two sets and U is the universal set such that n(U) = 800, n( A) = 250, n(B) = 350 and n(A ∩ B) = 100. Find n(A' ∩ B')

129 Views

148. S and T are two sets such that S ∪ T has 60 elements. S has 35 elements and S ∩ T has 7 elements. How many elements set T have ?

124 Views

149. In a school, there are 20 teachers who teach Mathematics or Physics or both. 12 teach Mathematics and 4 teach both Physics and Mathematics. How many teachers teach Physics?

142 Views

150. In a committee, 50 people speak French, 20 speak Spanish and 10 speak both French and Spanish. How many of them speak at least one of the two languages ? If there are a total of 75 members in the committee, how many of them neither speak French nor Spanish ?

149 Views