Tamilnadu Board-2018

State all the five properties of groups.

If $p$ is $T$ and $q$ is $F$, then which of the following have the truth value $T$?

Show that $p\to q \text{and} q\to p$ are not equivalent.

Prove that the identity element of a group is unique.

Prove that ${\left(a^{-1}\right)}^{-1}=a$ for every $a\in G$, group.

With usual notations, show that $\left(Z_n,{+}_n\right)$ forms a group.

Show that the set of all non-zero rational numbers is not closed under addition.

Show that $(p \wedge q) \to (p \vee q)$ is a tautology.