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Question Number 119797 by Ar Brandon last updated on 27/Oct/20

$$\mathrm{Q}1$$$$\mathrm{Let}{M}_2\mathrm{be}\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{square}\mathrm{matrices}\mathrm{of}\mathrm{order}2\mathrm{over}$$$$\mathrm{the}\mathrm{real}\mathrm{number}\mathrm{system}\mathrm{and}$$$$\mathcal{R}=\{(A,B)\in M_2\times M_2\mid A=P^{\mathrm{T}}BP\mathrm{for}\mathrm{some}$$$$\mathrm{non}-\mathrm{singular}P\in M\}$$$$\mathrm{Then}\mathcal{R}\mathrm{is}$$$$\left(\mathrm{A}\right)\mathrm{symmetric}$$$$\left(\mathrm{B}\right)\mathrm{transitive}$$$$\left(\mathrm{C}\right)\mathrm{reflexive}\mathrm{on}{M}_2$$$$\left(\mathrm{D}\right)\mathrm{not}\mathrm{an}\mathrm{equivalence}\mathrm{relation}\mathrm{on}{M}_2$$$$$$$$\mathrm{Q}2$$$$\mathrm{For}\mathrm{any}\mathrm{integer}n,\mathrm{let}{I}_n\mathrm{be}\mathrm{the}\mathrm{interval}(n,n+1).$$$$\mathrm{Define}$$$$\mathcal{R}=\{(\mathrm{x},\mathrm{y})\in \mathbb{R}\mid \mathrm{both}\mathrm{x},\mathrm{y}\in I_n\mathrm{for}\mathrm{some}n\in \mathbb{Z}\}$$$$\mathrm{Then}\mathcal{R}\mathrm{is}$$$$\left(\mathrm{A}\right)\mathrm{reflexive}\mathrm{on}\mathbb{R}$$$$\left(\mathrm{B}\right)\mathrm{symmetric}$$$$\left(\mathrm{C}\right)\mathrm{transitive}$$$$\left(\mathrm{D}\right)\mathrm{an}\mathrm{equivalence}\mathrm{relation}$$

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