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I have some problems understanding the notation used in this question.

Let $K:= \left\{P\in GL_{2}\mathbb{(R)}: P^{T}P=I_{2}\right\}, H:=\left\{A_{\theta}=\begin{pmatrix} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta) \end{pmatrix}: \theta\in \mathbb{R}\right\},$$ R = diag[1,-1]$, a diagonal matrix corresponding to reflection and $RH:=\left\{Rh: h\in H\right\}$ Show $K=H\biguplus RH$(disjoint union).

May I know how we interpret $H\biguplus RH$ ?

So, $x\in H\biguplus RH \Longleftrightarrow \exists h, g \in H$ such that $ x =hRg$ ?

Also, how we interpret $ \biguplus RH$ ? $z\in \biguplus RH \Longleftrightarrow \exists y\in RH$ such that $x\in y$ ? But any element $y\in RH$ is a matrix. What do we mean by, $x\in y$ ?

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No $H \uplus RH$ is short for disjoint union. So you have to show two things: (1) $H$ and $RH$ are disjoint, that is $H \cap RH = \emptyset$, (2) their union is $K$, that is $K = H \cup RH$.

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