Chapter 1: Relations and Functions

Ans: The given relation is \[\text{ }\!\!~\!\!\text{ R =  }\!\!\{\!\!\text{ }\left( \text{P, Q} \right)\text{ :}\] Distance of \[\text{P}\] from the origin is the same as the distance of \[\text{Q}\] from the origin}

Since, \[\left( \text{P, P} \right)\in \text{R}\]. 

The relation \[\text{R}\] is reflexive.

Let \[\left( \text{P, Q} \right)\in \text{R}\], distance of \[\text{P}\] from the origin is the same as the distance of \[\text{Q}\] from the origin similarly distance of \[\text{Q}\] from the origin will be the same as the distance of \[\text{P}\] from the origin. So, \[\left( \text{Q, P} \right)\in \text{R}\].

Therefore \[\text{R}\] is symmetric.

Let \[\left( \text{P, Q} \right),\left( \text{Q, S} \right)\in \text{R}\].

Distance of \[\text{P}\] from the origin is the same as the distance of \[\text{Q}\] from the origin and distance of \[\text{Q}\] from the origin is the same as the distance of \[\text{S}\] from the origin. So, the distance of \[\text{S}\] from the origin will be the same as the distance of \[\text{P}\] from the origin. So, \[\left( \text{P, S} \right)\in \text{R}\].

Therefore \[\text{R}\] is transitive.

Therefore the relation \[\text{R}\] is an equivalence relation.

The set of points related to \[\text{P}\ne \left( \text{0, 0} \right)\] will be those points whose distance from origin is same as distance of \[\text{P}\] from the origin and will form a circle with the centre as origin and this circle passes through \[\text{P}\].

Ans: The given relation is \[\text{R =  }\!\!\{\!\!\text{ }\left( {{\text{T}}_{\text{1}}}\text{, }{{\text{T}}_{\text{2}}} \right)\text{ : }{{\text{T}}_{\text{1}}}\] is similar to \[{{\text{T}}_{\text{2}}}\text{ }\!\!\}\!\!\text{ }\] since, every triangle is similar to itself.

The relation \[\text{R}\] is reflexive.

If \[\left( {{\text{T}}_{\text{1}}}\text{, }{{\text{T}}_{\text{2}}} \right)\in \text{R}\], then \[{{\text{T}}_{\text{1}}}\] is similar to \[{{\text{T}}_{\text{2}}}\]. 

\[\Rightarrow {{\text{T}}_{\text{2}}}\] is similar to \[{{\text{T}}_{\text{1}}}\].

\[\Rightarrow \left( {{\text{T}}_{\text{2}}}\text{, }{{\text{T}}_{\text{1}}} \right)\in \text{R}\]

Therefore \[\text{R}\] is symmetric.

Let \[\left( {{\text{T}}_{\text{1}}}\text{, }{{\text{T}}_{\text{2}}} \right)\text{,}\left( {{\text{T}}_{\text{2}}}\text{, }{{\text{T}}_{\text{3}}} \right)\in \text{R}\].

\[\Rightarrow {{\text{T}}_{\text{1}}}\] is similar to \[{{\text{T}}_{\text{2}}}\] and \[{{\text{T}}_{\text{2}}}\] is similar to \[{{\text{T}}_{\text{3}}}\].

\[\Rightarrow {{\text{T}}_{\text{1}}}\] is similar to \[{{\text{T}}_{\text{3}}}\].

\[\Rightarrow \left( {{\text{T}}_{\text{1}}}\text{, }{{\text{T}}_{\text{3}}} \right)\in \text{R}\]

Therefore \[\text{R}\] is transitive.

Therefore the relation \[\text{R}\] is an equivalence relation.

\[\dfrac{\text{3}}{\text{6}}\text{=}\dfrac{\text{4}}{\text{8}}\text{=}\dfrac{\text{5}}{\text{10}}\left( \text{=}\dfrac{\text{1}}{\text{2}} \right)\] 

Since, the corresponding sides of triangles \[{{\text{T}}_{\text{1}}}\] and \[{{\text{T}}_{\text{3}}}\] are in the same ratio, therefore triangle \[{{\text{T}}_{\text{1}}}\] is similar to triangle \[{{\text{T}}_{\text{3}}}\].

Hence, \[{{\text{T}}_{\text{1}}}\] is related to \[{{\text{T}}_{\text{3}}}\].

Ans: \[\text{R =  }\!\!\{\!\!\text{ }\left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{2}}} \right)\text{: }{{\text{P}}_{\text{1}}}\] and \[{{\text{P}}_{\text{2}}}\] have same number of sides\[\text{ }\!\!\}\!\!\text{ }\].

Since \[\left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{1}}} \right)\in \text{R}\] , as same polygon has same number of sides.

The relation \[\text{R}\] is reflexive.

Let \[\left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{2}}} \right)\in \text{R}\] .

\[\Rightarrow {{\text{P}}_{\text{1}}}\] and \[{{\text{P}}_{\text{2}}}\] have same number of sides.

\[\Rightarrow {{\text{P}}_{\text{2}}}\] and \[{{\text{P}}_{\text{1}}}\] have same number of sides.

\[\Rightarrow \left( {{\text{P}}_{\text{2}}}\text{, }{{\text{P}}_{\text{1}}} \right)\in \text{R}\]

Therefore \[\text{R}\] is symmetric.

Let \[\left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{2}}} \right)\text{,}\left( {{\text{P}}_{\text{2}}}\text{, }{{\text{P}}_{\text{3}}} \right)\in \text{R}\].

\[\Rightarrow {{\text{P}}_{\text{1}}}\] and \[{{\text{P}}_{\text{2}}}\] have same number of sides.

\[\Rightarrow {{\text{P}}_{\text{2}}}\] and \[{{\text{P}}_{\text{3}}}\] have same number of sides.

\[\Rightarrow {{\text{P}}_{\text{1}}}\] and \[{{\text{P}}_{\text{3}}}\] have same number of sides.

\[\Rightarrow \left( {{\text{P}}_{\text{1}}}\text{, }{{\text{P}}_{\text{3}}} \right)\in \text{R}\]

Therefore \[\text{R}\] is transitive.

Therefore the relation \[\text{R}\] is an equivalence relation.

The elements in $A$ related to right-angled triangle \[\left( \text{T} \right)\] with sides \[\text{3, 4}\] and \[\text{5}\] are the polygons having \[\text{3}\] sides.

Ans: \[\text{R =  }\!\!\{\!\!\text{ }\left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{2}}} \right)\text{: }{{\text{L}}_{\text{1}}}\] is parallel to \[{{\text{L}}_{\text{2}}}\text{ }\!\!\}\!\!\text{ }\]. 

The relation \[\text{R}\] is reflexive as any line \[{{\text{L}}_{\text{1}}}\] is parallel to itself, so, \[\left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{1}}} \right)\in \text{R}\].

Let \[\left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{2}}} \right)\in \text{R}\].

\[\Rightarrow {{\text{L}}_{\text{1}}}\] is parallel to \[{{\text{L}}_{\text{2}}}\], therefore \[{{\text{L}}_{\text{2}}}\] is parallel to \[{{\text{L}}_{\text{1}}}\].

\[\Rightarrow \left( {{\text{L}}_{\text{2}}}\text{, }{{\text{L}}_{\text{1}}} \right)\in \text{R}\]

Therefore \[\text{R}\] is symmetric.

Let \[\left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{2}}} \right)\text{,}\left( {{\text{L}}_{\text{2}}}\text{, }{{\text{L}}_{\text{3}}} \right)\in \text{R}\].

\[\Rightarrow {{\text{L}}_{\text{1}}}\] is parallel to \[{{\text{L}}_{\text{2}}}\]

\[\Rightarrow {{\text{L}}_{\text{2}}}\] is parallel to \[{{\text{L}}_{\text{3}}}\]

\[\Rightarrow {{\text{L}}_{\text{1}}}\] is parallel to \[{{\text{L}}_{\text{3}}}\]

\[\Rightarrow \left( {{\text{L}}_{\text{1}}}\text{, }{{\text{L}}_{\text{3}}} \right)\in \text{R}\]

Therefore \[\text{R}\] is transitive.

Therefore the relation \[\text{R}\] is an equivalence relation.

Set of all lines related to the line \[\text{y=2x+4}\] is the set of all lines that are parallel to the line \[\text{y=2x+4}\].

Slope of line \[\text{y=2x+4}\] is \[\text{m = 2}\]. Therefore, lines parallel to the given line are of the form \[\text{y=2x+c}\], where \[\text{c}\in \text{R}\].

Ans: \[\text{R = }\left\{ \left( \text{1, 2} \right)\text{, }\left( \text{2, 2} \right)\text{, }\left( \text{1, 1} \right)\text{, }\left( \text{4, 4} \right)\text{, }\left( \text{1, 3} \right)\text{, }\left( \text{3, 3} \right)\text{, }\left( \text{3, 2} \right) \right\}\].

Since \[\left( \text{a, a} \right)\in \text{R}\], for every \[\text{a}\in \left\{ \text{1, 2, 3, 4} \right\}\] 

The relation \[\text{R}\] is reflexive.

Since \[\left( \text{1, 2} \right)\in \text{R}\] , but \[\left( \text{2, 1} \right)\notin \text{R}\] .

Therefore \[\text{R}\] is not symmetric.

\[\left( \text{a, b} \right)\text{,}\left( \text{b, c} \right)\in \text{R}\Rightarrow \left( \text{a, c} \right)\in \text{R}\] for all \[\text{a, b, c}\in \left\{ \text{1, 2, 3, 4} \right\}\].

Therefore \[\text{R}\] is transitive.

Therefore the relation \[\text{R}\] is reflexive and transitive but not symmetric.

The correct answer is (\[B\]) \[\text{R}\] is reflexive and transitive but not symmetric.

Ans: The given relation is \[\text{R = }\left\{ \left( \text{a, b} \right)\text{: a = b - 2, b > 6} \right\}\]

Now,

Considering \[\left( \text{2, 4} \right)\in \text{R}\].

Since, \[\text{b  6}\], so, \[\left( \text{2, 4} \right)\notin \text{R}\].

Considering \[\left( \text{3, 8} \right)\in \text{R}\].

Since \[\text{3 }\ne \text{ 8 - 2}\], so \[\left( \text{3, 8} \right)\notin \text{R}\].

Considering \[\left( \text{6, 8} \right)\in \text{R}\]. 

Since \[\text{86}\] and \[\text{6=8-2}\], so \[\left( \text{6, 8} \right)\in \text{R}\].

Therefore, the correct answer is ($C$)\[\left( \text{6, 8} \right)\in \text{R}\].