Question

Check whether the relation $R$ in $R$ defined by $R=\{(a,b):a\le b^3\}$ is reflexive, symmetric or transitive.

Answer 1

$R=\{(a,b):a\le b^3\}$

Here $R$ is set of real numbers

Hence both $a$ and $b$ are real numbers.

$1.$If the relation is reflexive, then $(a,a)\in R$

i.e.,$a\le a^3$

For $a=1,a^3\le 1\Rightarrow 1\le 1$

For $a=2,a^3\le 8\Rightarrow 2\le 8$

For $a=\frac{1}{2},a^3\ge \frac{1}{8}$

Hence $a\le a^3$ is not true for all values of $a$

So, the given relation is not relexive.

To check whether symmetric or not:

If $(a,b)\in R$ then $(b,a)\in R$

If $a\le b^3$ then $b\le a^3$

For $a=2,b=3,2<2^3,3<2^3$

For $a=2,b=9,2<9^3,9>2^3$

Since $b\le a^3$ is not true for all values of $a$ and $b$

Hence the given relation is not symmetric.

To check whether transitive or not:

If $(a,b)\in R$ and $(b,c)\in R$ then $(a,c)\in R$

If $a\le b^3$ and $b\le c^3$ then $a\le c^3$

For $a=1,b=2,c=3,b^3=8,c^3=27\Rightarrow a\le b^3,b\le c^3$ and $a\le c^3$

For $a=3,b=\frac{3}{2},c=\frac{4}{3},b^3={\left(\frac{3}{2}\right)}^3=3.375,c^3={\left(\frac{4}{3}\right)}^3=2.37\Rightarrow a\le b^3,b\le c^3$ and $a\ge c^3$

Since if $a\le b^3,b\le c^3$ and $a\le c^3$ is not true for all values of $a,b,c$.

Hence, the given relation is not transitive.

$\therefore$ the given relation is neither reflexive, symmetric or transitive.