Answer
Verified
416.7k+ views
Hint: Here we have been given that $n$ is a factor of $m$. So use $(a,a)\in R\forall a\in A$, $\forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc$ and $(a,b)\in R\Rightarrow (b,a)\in R$. You will get the answer.
Complete step-by-step answer:
In maths, a binary relation $R$ across a set $X$ is reflexive if each element of set $X$ is related or linked to itself. In terms of relations, this can be defined with $(a,a)\in R\forall a\in X$ or $I\in R$ where I is the identity relation on A. It has a reflexive property and is said to hold reflexivity. Symmetry, transitivity and reflexivity are the three properties representing equivalence relations.
In relation and functions, a reflexive relation is the one in which every element maps to itself. For example, let us consider a set $A=\{1,2\}$. Now here the reflexive relation will be $R=\{(1,1),(2,2),(1,2),(2.1)\}$. Hence, a relation is reflexive if:
$(a,a)\in R\forall a\in A$
where $a$ is the element, $A$ is the set and $R$ is the relation.
A binary relation $R$ over a set $X$ is transitive if whenever an element $a$ is related to an element $b$, and $b$ is in turn related to an element $c$, then a is also related to $c$.
$\forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc$
On the other hand, "is the mother of" is not a transitive relation, because if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire. What is more, it is anti transitive: Alice can never be the mother of Claire.
Let $A$ be a set in which the relation $R$ defined. Then $R$ is said to be a symmetric relation, if $(a,b)\in R\Rightarrow (b,a)\in R$, that is, $aRb\Rightarrow bRa$ for all $(a,b)\in R$.
Consider, for example, the set $A$ of natural numbers. If a relation $A$ be defined by $''(x+y)''$, then this relation is symmetric in $A$, for $a+b=5\Rightarrow b+a=5$.
But in the set $A$ of natural numbers if the relation $R$ be defined as ‘$x$ is a divisor of $y$’, then the relation $R$ is not symmetric as $3R9$ does not imply $9R3$; for, $3$ divides $9$ but $9$ does not divide $3$.
For a symmetric relation $R$, ${{R}^{-1}}=R$.
Since $n$ is a factor of $n$, since every natural number is a factor of itself so the relation is reflexive.
If $n$ is a factor of $m$ and $m$ is a factor of $p$, then $n$ is surely a factor of $p$, so the relation is transitive.
If however $n$ is a factor of $m$, $m$ is not necessarily a factor of $n$ so the relation is not symmetric.
Hence, the answer is option D.
Note: Read the question carefully. You should know the concept of reflexive, transitive and symmetric. Also, you should know the basics of these and their properties. You must also know the types of properties.
Complete step-by-step answer:
In maths, a binary relation $R$ across a set $X$ is reflexive if each element of set $X$ is related or linked to itself. In terms of relations, this can be defined with $(a,a)\in R\forall a\in X$ or $I\in R$ where I is the identity relation on A. It has a reflexive property and is said to hold reflexivity. Symmetry, transitivity and reflexivity are the three properties representing equivalence relations.
In relation and functions, a reflexive relation is the one in which every element maps to itself. For example, let us consider a set $A=\{1,2\}$. Now here the reflexive relation will be $R=\{(1,1),(2,2),(1,2),(2.1)\}$. Hence, a relation is reflexive if:
$(a,a)\in R\forall a\in A$
where $a$ is the element, $A$ is the set and $R$ is the relation.
A binary relation $R$ over a set $X$ is transitive if whenever an element $a$ is related to an element $b$, and $b$ is in turn related to an element $c$, then a is also related to $c$.
$\forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc$
On the other hand, "is the mother of" is not a transitive relation, because if Alice is the mother of Brenda, and Brenda is the mother of Claire, then Alice is not the mother of Claire. What is more, it is anti transitive: Alice can never be the mother of Claire.
Let $A$ be a set in which the relation $R$ defined. Then $R$ is said to be a symmetric relation, if $(a,b)\in R\Rightarrow (b,a)\in R$, that is, $aRb\Rightarrow bRa$ for all $(a,b)\in R$.
Consider, for example, the set $A$ of natural numbers. If a relation $A$ be defined by $''(x+y)''$, then this relation is symmetric in $A$, for $a+b=5\Rightarrow b+a=5$.
But in the set $A$ of natural numbers if the relation $R$ be defined as ‘$x$ is a divisor of $y$’, then the relation $R$ is not symmetric as $3R9$ does not imply $9R3$; for, $3$ divides $9$ but $9$ does not divide $3$.
For a symmetric relation $R$, ${{R}^{-1}}=R$.
Since $n$ is a factor of $n$, since every natural number is a factor of itself so the relation is reflexive.
If $n$ is a factor of $m$ and $m$ is a factor of $p$, then $n$ is surely a factor of $p$, so the relation is transitive.
If however $n$ is a factor of $m$, $m$ is not necessarily a factor of $n$ so the relation is not symmetric.
Hence, the answer is option D.
Note: Read the question carefully. You should know the concept of reflexive, transitive and symmetric. Also, you should know the basics of these and their properties. You must also know the types of properties.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE