Let R be relation defined in set A={1,2,3,4,5,6,7} by R ={(a, b): both a and b are either odd

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1.Relation and Function

easy

Let $R$ be the relation defined in the set $A=\{1,2,3,4,5,6,7\}$ by $R =\{(a, b):$ both $a$ and $b$ are either odd or even $\} .$ Show that $R$ is an equivalence relation. Further, show that all the elements of the subset $ \{1,3,5,7\}$ are related to each other and all the elements of the subset $\{2,4,6\}$ are related to each other, but no element of the subset $\{1,3,5,7\}$ is related to any element of the subset $\{2,4,6\} .$

Option A

Option B

Option C

Option D

Solution

Given any element $a$ in $A$, both $a$ and $a$ must be either odd or even, so that $(a, a) \in R$ Further, $(a, \,b) \in R $ $\Rightarrow$ both $a$ and $b$ must be either odd or even $\Rightarrow(b, a) \in$ $R$ similarly, $(a,\, b) \in R$ and $(b,\, c) \in R$ $\Rightarrow$ all elements $a, \,b,\, c,$ must be either even or odd simultaneously $\Rightarrow(a, \,c) \in R$. Hence, $R$ is an equivalence relation. Further, all the elements of $\{1,3,5,7\}$ are related to each other, as all the elements of this subset are odd. Similarly, all the elements of the subset $ \{2,4,6\} $ are related to each other, as all of them are even. Also, no element of the subset $\{1,3,5,7\}$ can be related to any element of $\{2,4,6\}$ , as elements of $\{1,3,5,7\}$ are odd, while elements of $\{2,4,6\}$ are even.

Standard 12

Mathematics

Let a relation $R$ be defined by $R = \{(4, 5); (1, 4); (4, 6); (7, 6); (3, 7)\}$ then ${R^{ – 1}}oR$ is

medium

Let $L$ denote the set of all straight lines in a plane. Let a relation $R$ be defined by $\alpha R\beta \Leftrightarrow \alpha \bot \beta ,\,\alpha ,\,\beta \in L$. Then $R$ is

easy

Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(a,b) \, | a,b \in R$ and $a – b + \sqrt 3$ is an irrational number$\}$ The relation $r$ is

normal

Given a non empty set $X$, consider $P ( X )$ which is the set of all subsets of $X$.

Define the relation $R$ in $P(X)$ as follows :

For subsets $A,\, B$ in $P(X),$ $ARB$ if and only if $A \subset B .$ Is $R$ an equivalence relation on $P ( X ) ?$ Justify your answer.

hard

Check whether the relation $R$ defined in the set $\{1,2,3,4,5,6\}$ as $R =\{(a, b): b=a+1\}$ is reflexive, symmetric or transitive.