Are these two statements true or false?

Question

1. For all sets $A,B$ we have that $|A \cup B|=|A|+|B|$ whereby $|\text{ }|$ stands for the cardinality.

2. For all vectors $v,w \in \mathbb{R}^{2}$ the vector $u= \left \langle v,w \right \rangle v \text{ } - \left\|v\right\|^{2}w$ is located vertically at $v$ whereby $\left \langle , \right \rangle$ stands for the euclidean scalar product on $\mathbb{R}^{2}$

I think first statement is false because to be true it must be subtracted by $|A \cap B|$ and also this must be finite and it's not stated in the task either. Or that is wrong? I still say it's false because this is unclear from task still.

About the other statement I have no idea at all : /

Answer 1

Your reasoning for the first statement is completely correct.

For the second statement, I believe "located vertically at $v$" means "orthogonal to $v$." To prove that two vectors are orthogonal, take their dot product and show that it is $0$. In other words, show that: $$\langle u, v \rangle=\langle \langle v, w \rangle v-\|v\|^2w, v\rangle$$ is equal to $0$. (Hint: Remember that $\langle v, v \rangle=\| v \|^2$.) Good luck!

Answer 2

If $$ u= \left \langle v,w \right \rangle v \text{ } - \left\|v\right\|^{2}w $$ then \begin{align} \langle u, v \rangle &= \left \langle \left \langle v,w \right \rangle v \text{ } - \left\|v\right\|^{2}w, v \right\rangle \\ &= \left \langle \left \langle v,w \right \rangle v, v \right\rangle \text{ } - \left \langle \left\|v\right\|^{2}w, v \right\rangle \\ &= \left \langle v,w \right \rangle \left \langle v, v \right\rangle \text{ } - \left\|v\right\|^{2} \left \langle w, v \right\rangle \\ &= \left \langle v,w \right \rangle \left\|v\right\|^{2} \text{ } - \left\|v\right\|^{2} \left \langle w, v \right\rangle \\ &=0 \end{align} which means $v \perp w$.