Which of the following is false?

$|x+y|=\left|x\right|+\left|y\right|\Rightarrow x\text{ and }y\text{ have same sign or at least one of x and y is zero.}$

$\Rightarrow (x,y)\text{ lies in }{1}^{\text{st }}\text{ or }{3}^{\text{rd }}\text{ quadrant or any of the }x\text{-axis or y-axis}$

$|x+y|<\left|x\right|+\left|y\right|\Rightarrow x\text{ and }y\text{ have opposite signs }\Rightarrow (x,y)\text{ lies in }{2}^{\text{nd }}\text{ or }{4}^{\text{th }}\text{ quadrant }$

$|x-y|=\left|x\right|+\left|y\right|\Rightarrow x\text{ and }-y\text{ have same }\mathrm{sign}\Rightarrow x\text{ and }y\text{ have opposite signs }\Rightarrow (x,y)\text{ lies in }{2}^{\text{nd }}\text{ or }{4}^{\text{th }}\text{ quadrant }$