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A relation $\text{R}$ on a set $\text{A}$ is said to be Total Relation iff $a\text{R}b$ Or $b\text{R}a$ Or both, for all $a,b \in \mathrm{A}$.

Which of the following options is/are false?

Which of the following options is/are false?

- Every Total relation is reflexive.
- If A relation $\text{S}$ is total and symmetric then $\text{S}$ is an equivalence relation.
- If a relation $\mathrm{S}$ is total and transitive, then $\mathrm{S}$ is an equivalence relation.
- The number of total relations on a set of $5$ elements is $1024.$

4 votes

By definition of total relation, for every two elements $a,b$(same or different), $a\text{R}b$ Or $b\text{R}a$ Or both. So, every total relation is reflexive.

If $\text{A}$ relation $\text{S}$ is total and symmetric then relation $\text{S}= \text{A} \ast \text{A}$, i.e. $\text{S}$ is a universal relation on $\mathrm{A}$. $\mathrm{So}, \mathrm{S}$ is an equivalence relation.

The number of total relations on a set of $5$ elements is $3^{10}=59,049$.

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