Irreflexive → YES.

Symmetric → NO.

Antisymmetric → NO.

Transitive → NO.

Asymmetric → NO.

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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**?

- $R$ is symmetric but NOT antisymmetric
- $R$ is NOT symmetric but antisymmetric
- $R$ is both symmetric and antisymmetric
- $R$ is neither symmetric nor antisymmetric

Best answer

**Answer**** is D. **

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.

**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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