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Question

A relation on a set $A$ is said to be symmetric iff $\left( {a,b} \right) \in A \Rightarrow \left( {b,a} \right) \in A,\forall a,b \in A$

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Vedclass · Accepted Answer

$2^{15}$

Vedclass · Answer

A relation on a set $A$ is said to be symmetric iff $\left( {a,b} ight) \in A \Rightarrow \left( {b,a} ight) \in A,\forall a,b \in A$

Here $A = \left\{ {3,4,6,8,9} ight\}$

Number of order paris of $A 	imes A = 5 	imes 5 = 25$

Divide $25$ order pairs of $A 	imes A$ in 3 parts as follows:

Part- $A:\left( {3,3} ight),\left( {4,4} ight),\left( {6,6} ight),\left( {8,8} ight),\left( {9,9} ight)$

Part- $B:\left( {3,4} ight),\left( {3,6} ight),\left( {3,8} ight),\left( {3,9} ight),\left( {4,6} ight),\left( {4,8} ight),\left( {4,9} ight),\left( {6,8} ight),\left( {6,9} ight),\left( {8,9} ight)$

Part- $C:\left( {4,3} ight),\left( {6,3} ight),\left( {8,3} ight),\left( {9,3} ight),\left( {6,4} ight),\left( {8,4} ight),\left( {9,4} ight),\left( {8,6} ight),\left( {9,6} ight),\left( {9,8} ight)$

In part - $A$ , both components of each order pair are same.

In part - $B$ , both components are different but not two such order paris are present in which fiest component of one order pair is the second component of another order pair and vice-versa.

In part - $C$ , only reverse of the order pairs of part - $B$ are peresent i.e., if $(a,b)$ is persent in part - $B$, then $(b,a)$ will be present in part - $C$ For example $(4,3)$ is peresnt in part - $C$ .

Number of order pair in $A$,$B$ and $C$ are $5,10$ and $10$ respectively.

In any symmetic relation on set $A$, if any order pair of part - $B$ is present then its reverse order pair of part - $C$ will must be also present.

Hence number of symmetric relation on set $A$ is equal to the number of all relations on aset $D$, which contains all the order pairs of part - $A$ and part - $B$.

Now $n\left( D ight) = n\left( A ight) + n\left( B ight) = 5 + 10 = 15$

Hence number of all relations on set $D = {\left( 2 ight)^{15}}$

$ \Rightarrow $ Number of symmetric relations on set $D = {\left( 2 ight)^{15}}$

If $A = \left\{ {x \in {z^ + }\,:x < 10} \right.$& and $x$ is a multiple of $3$ or $4\}$, where $z^+$ is the set of positive integers, then the total number of symmetric relations on $A$ is

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