D.P. GUPTA, Preetima Bajpai and, Sanjeev Kumar Jha Solutions for Chapter: Mathematical Reasoning, Binary Operation and Group Theory, Exercise 2: Exercise 2

The identity element for the binary operation $*$ defined by $a^{*} b=\frac{ab}{2}, a, b\in Q_0$ (the set of all non-zero rational numbers) is

The set of all non-zero real number with the operation defined on it by a ${}^{*} b=\frac{ab}{2}$ is an abelian group. The identity of the group is

If $p, q, r$ are simple propositions with truth values $T, F, T$ then the truth value of $(~p\vee q)\wedge ~r\Rightarrow p$ is

In the group of non-zero rational numbers under the binary operation $*$ given by  $a{}^{*} b=\frac{ab}{5}$ the identity element and the inverse of $8$ are respectively.

If $\left(G{,}^{*}\right)$ is a group such that $(a* b{)}^2={\left(a^{*}a\right)}^{*}\left(b^{*} b\right)$ for all $a, b, \in G,$ then $G$ is

In a group $\left(G,*\right)$ if there exists an element $b$ such that $b^{*}a=b$ for all $a\in G\mathit{,}$ then order of the group is

If $G$ is a group of even order then

The logical expression $X,$ in its simplest form for the truth table