Properties Of The Less Than Relation

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Problem 54: (Properties Of The Less Than Relation)

Let $A=\mathbb{R}$. Given $a,b\in \mathbb{R}$, let's say that $a$ and $b$ are related by $\mathrm{R}$ if and only if $a$ is less than $b$. So using symbols we can write $a\mathrm{R}b$ if and only if $a<b$.

1. Is $R$ reflexive? Remember to justify your answer, here and below.
2. Is $R$ symmetric?
3. Is $R$ transitive?
4. Is $R$ antisymmetric?

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