Question 136
a) Let $H_{1}$ and $H_{2}$ be subgroups of the group $(G, \cdot)$. Show that $H_{1} \cap H_{2}$ is a subgroup of the group $(G, \cdot)$.
b) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a bijective function with $f^{-1}(1)=2$, determine the neutral element of the composition law defined by:
$$
x * y = f\left(f^{-1}(x) + f^{-1}(y) - 2\right), \forall x, y \in \mathbb{R}
$$
b) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a bijective function with $f^{-1}(1)=2$, determine the neutral element of the composition law defined by:
$$
x * y = f\left(f^{-1}(x) + f^{-1}(y) - 2\right), \forall x, y \in \mathbb{R}
$$