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Consider the binary relation $R = \left\{(x,y), (x,z), (z,x), (z,y)\right\}$ on the set $\{x,y,z\}$. Which one of the following is TRUE?

1. $R$ is symmetric but NOT antisymmetric
2. $R$ is NOT symmetric but antisymmetric
3. $R$ is both symmetric and antisymmetric
4. $R$ is neither symmetric nor antisymmetric
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A binary relation $R$ over a set $X$ is symmetric if it holds for all $a$ and $b$ in $X$ that if $a$ is related to $a.$

$\forall_{a,b} \in X,aRb \Rightarrow bRa.$

Here $(x,y)$ is there in $R$ but $(y,x)$ is not there.

$\therefore$ Not Symmetric.

For Antisymmetric Relations: $\forall_{a,b} \in X, R(a,b) \;\& \;R(b,a)\Rightarrow a=b.$

Here $(x,z)$ is there in $R$ also $(z,x)$ is there violating the antisymmetric rule.

$\therefore$ Not AntiSymmetric.

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Draw digraph and check

For symmetry, for every one directional edge, there must be other edge in reverse direction.

For Anti-Symmetry, only way arrows are allowed except self loops.

So, not Symmetric and not Anti Symmetric

For Symmetric property:   If xRy then yRx for all x,y∈ set {x,y,z}

But in this relation xRy but not yRx . So,it is not symmetric relation.

For Antisymmetric property:   If xRy and yRx then x = y ,for all x,y∈ set {x,y,z}

But in this relation xRz and zRx  but not x = z ,so,it is not antisymmetric relation.

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