Prove the following equivalencies.

$($a$)HZH=x$

$($b$)HxH=Z$

$($c$)HYH=-Y$

$($d$)CNO{T}_{1,0}=H^{ox2}CNO{T}_{0,1}{H}^{ox2}$

In $($d$),$ CNOT ${?}_{i,j}$ means a CNOT with qubit $i$ as the control and qubit $j$ as the

target. $H^{ox2}$ means a Hadamard gate applied to both qubits.