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Relations and Functions

Short Answer Type

41.

Let R be a relation on set N of natural numbers defined by R = {(a, b) : a, b ∊ N, a + 3b = 12} (or a R b : a + 3b = 12).

Find (i) R in Roster form (ii) Domain of R (iii) Range of R.

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42. Let A = {1, 3, 5}, B = {7,11}. If R = {(a, b) : a ∊ A, b e B, a – b = odd}, then show that R is an empty relation.

Here, A = {1, 3, 5}, B = {7, 11},

$space space space space space space space space space space straight R space equals space left curly bracket left parenthesis straight a comma space straight b right parenthesis space colon space straight a element of straight A comma space straight b element of straight B comma space straight a minus straight b space is space odd right curly bracket$

a-b = 1 - 7 = -6 is not odd, 1 - 11 = -10 is not odd, 3-7= -4 is not odd.

3-11 = -8 is not odd, 5-7 = -2 is not odd, 5 - 11 = -6 is not odd.

$rightwards double arrow$ $space space space space space space space space space space space space space space space space space space space space space space space space space straight R space equals space space straight ϕ$

Hence, R is an empty relation.

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43. Determine the domain and range of a relation R defined by :

R = {(x + 1), (x + 5) : x = 0, 1, 2, 3, 4, 5}

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44.

Let R be the relation on Z defined by R = {(a,b): a, b ∈ Z, a – b is an integer}. Find the domain and range of R.

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45. Let R be a relation from N to N, defined by

space space space space space straight R space equals space left curly bracket left parenthesis straight a comma space straight b right parenthesis space colon space straight a comma space straight b space element of straight N space and space straight a space equals space straight b squared right curly bracket

. Are the following true?

left parenthesis straight i right parenthesis space left parenthesis straight a comma space straight a right parenthesis element of straight R space for space all space straight a element of straight N space

left parenthesis ii right parenthesis space left parenthesis straight a comma space straight b right parenthesis element of straight R rightwards double arrow left parenthesis straight b comma straight a right parenthesis element of straight R

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108 Views

46.

Let R be a relation from Q to Q defined by :

$space space space space space space space space space space space space space space space space straight R space equals space left curly bracket left parenthesis straight a comma space straight b right parenthesis space colon straight a element of straight Q comma space straight b element of straight Q comma space straight a minus straight b element of straight Z$

Show that

$space space space space space space left parenthesis straight i right parenthesis space left parenthesis straight a comma space straight a right parenthesis element of straight R for all straight a element of straight Q$

$space space space space space space left parenthesis ii right parenthesis space left parenthesis straight a comma space straight b right parenthesis element of straight R rightwards double arrow left parenthesis straight b comma straight a right parenthesis element of straight R$

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47.

Determine the domain and range of relation :

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

(b) R = $space space space space left curly bracket left parenthesis straight x comma space straight y right parenthesis space colon space straight x element of straight N comma space 3 less than straight x less than 7 comma space straight y space equals 3 right curly bracket$