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

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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 $3^{10}=59,049$.

Video Solution:

https://youtu.be/tqjuxfutFHg?t=142

is the number of total relations calculated like this:

relations in 5 elements is 10 by $\frac {n(n-1)}2$ and each one has 3 possible ways either aRb or bRa or both so $3^{10}$

@Deepak Poonia

Watch Video Solution here:

https://youtu.be/tqjuxfutFHg?t=142