Answer
Verified
452.1k+ views
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 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.
Complete step by step answer:
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE