let S be the set of all real numbers and let R be a relation in S, defined by \[R=\left\{ \left( a,b \right):a\le {{b}^{3}} \right\}\]. Show that R satisfies none of reflexivity, symmetry and transitivity.

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let S be the set of all real numbers and let R be a relation in S, defined by \[R=\left\{ \left( a,b \right):a\le {{b}^{3}} \right\}\]. Show that R satisfies none of reflexivity, symmetry and transitivity.

Vedantu Content Team · Accepted Answer

Hint: To show that R is not reflexive on S, find an example such that $\left( a,a \right)\in R$ for $a\in S$. To show that R is not symmetric, find an example such that $\left( a,b \right)\in R$ and $\left( b,a \right)\notin R$ for $a,b\in S$. To show that ‘R’ is not transitive, find an example such that $\left( a,b \right)\in R$ and $\left( b,c \right)\in R$ and $\left( a,c \right)\notin R$ for $a,b,c\in S$.Complete step by step answer:Let ‘A’ be set thenReflexivity, symmetry and transitivity of a relation on set ‘A’ is defined as follows.Reflexive relation: A relation R on a set ‘A’ is said to be reflexive if every element of A is related to itself i.e. for every element say (a) in set A, $\left( a,a \right)\in R$.Thus, R on a set A is not reflexive if there exists an element $a\in A$ such that $\left( a,a \right)\in R$.Symmetric Relation: A relation R on a set ‘A’ is said to be symmetric relation if $\left( a,b \right)\in R$ then (b, a) must belong to R i.e. $\left( a,b \right)\in R\Rightarrow \left( b,a \right)\in R\ for\ all\ a,b\in A$.Transitive Relation: A relation R on A is said to be transitive relation if $\left( a,b \right)\in R$ and $\left( b,c \right)\in R$ then $\left( a,c \right)\in R$ i.e. $$\left( a,b \right)\in R\ and\ \left( b,c \right)\in R\Rightarrow \left( a,c \right)\in R$$.Let us check one by one.Let us first check for reflexivity.To show that R does not satisfy reflexivity, we have to find at least one element in ‘S’ such that $\left( a,a \right)\in R$.Let us take $a=-2$  (‘-2’ is a real number so, $a\in S$)${{a}^{3}}={{\left( -2 \right)}^{3}}=-8$We know, $-8For $\left( a,b \right)$ to belong to R, a should be less than or equal to $${{b}^{3}}$$i.e. $$a\le {{b}^{3}}$$As $-2\ge -8,\left( -2,-8 \right)\notin R$So, for $a=-2,\left( a,a \right)\notin R$This, R doesn’t satisfy reflexivity on set ‘S’.Now, let us check for symmetry.We have to show that R doesn’t satisfy symmetry.To show that R doesn’t satisfy symmetry, we have to find at least one example such that  $\left( a,b \right)\in R$ and $\left( b,a \right)\notin R$.We have to show that for any $a,b\in S,\left( a,b \right)\in R$ but $\left( b,a \right)\notin R$Let a = 1 and b = 2 both ‘1’ and ‘2’ are real numbers.So $a\in S\ and\ b\in S$For $\left( 1,2 \right)\to 1\le {{\left( 2 \right)}^{3}}\ so,\ \left( 1,2 \right)\in R$But for $\left( 2,1 \right)\to 2>{{\left( 1 \right)}^{3}}\ so,\ \left( 2,1 \right)\notin R$Thus, we have got example in the set ‘S’ which doesn’t symmetry on set S.So ‘R’ doesn’t satisfy symmetry on set S.Now let us check for transitivity.We have to show that R doesn’t satisfy transitivity on S. To show that R doesn’t satisfy on S, we have to find an example such $\left( a,b \right)\in R\ and\ \left( b,c \right)\in R\ but\ \left( a,c \right)\notin R$.Let us take a = 63, b = 4 and  c = 2As $$63\le {{\left( 4 \right)}^{3}}\ i.e.\ a\le {{\left( b \right)}^{3}}\ so,\ \left( a,b \right)\in R$$And $$4\le {{\left( 2 \right)}^{3}}\ i.e.\ b\le {{\left( c \right)}^{3}}\ so,\ \left( b,c \right)\in R$$But $$63>{{\left( 2 \right)}^{3}}\ i.e.\ a>{{c}^{3}}\ so,\ \left( a,c \right)\notin R$$Thus we have got an example in S which doesn’t satisfy transitivity on set S.So, R doesn’t satisfy transitivity on set ‘S’.Hence, we have shown that R satisfies none of reflexivity, symmetry and transitivity on S.Note: To show that R doesn’t satisfy reflexivity, symmetry and transitivity, we don’t need to prove that $a > {{b}^{3}}$ for all values of set S or for an interval of values of set S. We just need to find at least one example for each which doesn’t satisfy reflexivity, symmetry and transitivity.

let S be the set of all real numbers and let R be a relation in S, defined by \[R=\left\{ \left( a,b \right):a\le {{b}^{3}} \right\}\]. Show that R satisfies none of reflexivity, symmetry and transitivity.

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let S be the set of all real numbers and let R be a relation in S, defined by \[R=\left\{ \left( a,b \right):a\le {{b}^{3}} \right\}\]. Show that R satisfies none of reflexivity, symmetry and transitivity.

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