# GATE Papers >> CSE >> 2019 >> Question No 16

Question No. 16

Which one of the following is NOT a valid identity?

##### Answer : (B) $(\mathrm x+\mathrm y)\oplus\mathrm z\;=\;\mathrm x\oplus(\mathrm y+\mathrm z)$

Solution of Question No 16 of GATE 2019 CSE Paper

$\begin{array}{l}\left(\mathrm A\right)\mathrm L\;\mathrm H\;\mathrm S=\mathrm x\oplus\mathrm y\\\;\;\;\;\;\mathrm R\;\mathrm H\;\mathrm S=\left(\mathrm{xy}+\mathrm x'\mathrm y'\right)'=\left(\mathrm x\odot\mathrm y'\right)=\mathrm x\oplus\mathrm y\\\;\;\;\;\;\mathrm L\;\mathrm H\;\mathrm S=\mathrm R\;\mathrm H\;\mathrm S\Rightarrow\mathrm{true}\\\left(\mathrm B\right)\mathrm L\;\mathrm H\;\mathrm S=\mathrm x\oplus\mathrm y;\;\mathrm{condition}\;\mathrm x.\mathrm y=0;\;\mathrm R\;\mathrm H\;\mathrm S=\mathrm x+\mathrm y\end{array}$

 x y L H S R H S 0 0 0 0 0 1 1 1 1 0 1 1

⇒ True

$\begin{array}{l}\left(\mathrm C\right)\mathrm L\;\mathrm H\;\mathrm S=(\mathrm x+\mathrm y)\oplus\mathrm z\\\;\;\;\;\;\mathrm R\;\mathrm H\;\mathrm S=\mathrm x\oplus\left(\mathrm y+\mathrm z\right)\end{array}$

 X y z L H S R H S 0 1 1 0 1

$\begin{array}{l}\;\;\;\;\;\;\mathrm L\;\mathrm H\;\mathrm S\neq\mathrm R\;\mathrm H\;\mathrm S\Rightarrow\mathrm{False}\\\left(\mathrm D\right)\mathrm L\;\mathrm H\;\mathrm S=(\mathrm x\oplus\mathrm y)\oplus\mathrm z\\\;\;\;\;\;\mathrm R\;\mathrm H\;\mathrm S=\mathrm x\oplus\left(\mathrm y\oplus\mathrm z\right)\end{array}$

 X y z L H S R H S 0 1 1 0 0

As associative ⇒ True

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