Question

# 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.

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 $a,b,c\in S$.

Let ‘A’ be set then
Reflexivity, 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, $-8<-2$
For $\left( a,b \right)$ 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 = 2
As $63\le {{\left( 4 \right)}^{3}}\ i.e.\ a\le {{\left( b \right)}^{3}}\ so,\ \left( a,b \right)\in R$
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.