Let A=-2,-1,0,1,2,3.

Question

Let $A={-2,-1,0,1,2,3}$. Let R be a relation on $A$ defined by $xRy$ if and only if $y=max{x,1}$. Let $l$ be the number of elements in R . Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l+m+n$ is equal to

Accepted Answer

To solve the problem, we start by defining the set $A={-2,-1,0,1,2,3}$ and the relation $R$ on set $A$, where an element $x$ is related to $y$ (written as $x\, R\, y$) if and only if $y=max{x,1}$.

This leads us to the following pairs in the relation $R$:

For $x=-2$, $y=max{-2,1}=1$, so $(-2,1)$ is in $R$.

For $x=-1$, $y=max{-1,1}=1$, so $(-1,1)$ is in $R$.

For $x=0$, $y=max{0,1}=1$, so $(0,1)$ is in $R$.

For $x=1$, $y=max{1,1}=1$, so $(1,1)$ is in $R$.

For $x=2$, $y=max{2,1}=2$, so $(2,2)$ is in $R$.

For $x=3$, $y=max{3,1}=3$, so $(3,3)$ is in $R$.

Thus, the relation $R$ consists of the pairs: ${(-2,1),(-1,1),(0,1),(1,1),(2,2),(3,3)}$, and there are $l=6$ elements in $R$.

Making the Relation Reflexive
A relation is reflexive if every element in the set $A$ relates to itself. Therefore, the missing reflexive pairs are:

$(-2,-2)$

$(-1,-1)$

$(0,0)$

Adding these three pairs will make the relation reflexive, so $m=3$.

Making the Relation Symmetric
A relation is symmetric if whenever $(x,y)$ is in $R$, $(y,x)$ must also be in $R$. Therefore, the missing symmetric pairs are:

$(1,-2)$

$(1,-1)$

$(1,0)$

Thus, we need to add these three pairs for symmetry, so $n=3$.

Finally, we calculate the sum $l+m+n=6+3+3=12$.

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