Element choice for equivalence relation

Question

Assume that $T = \{v,w,x,y,z\}$ and that $w \sim x \;$ and $x \sim y$. Which of the following(s) must also be true if $\sim$ is to be an equivalence relation?

$$(a)\;z\sim z,\;\;(b)\;x\sim w,\;\;(c)\;v \sim z,\;\;(d)\;y \sim w$$

My Solution :

$(b)$ must be true since;

$$\text{If}\;\; w\sim x,\; \text{then}\; x\sim w$$

And I think $(d)$ must also be true because;

There are $w\sim x$ and $x \sim y$ so there is $w \sim y$. That is,

$$\text{If}\;\; w\sim y,\; \text{then}\; y\sim w$$

Is it true? Thanks in advance!

Answer 1

Since $\sim$ is an equivalence on $T$, then the following are right:

$$ \begin{array}{ll} v\sim v, w\sim w, x\sim x,y\sim y, z\sim z&\text{by reflexivity}\\ w\sim y&\text{by transitivity}\\ x\sim w, y\sim x, y\sim w&\text{by symmetry}. \end{array} $$

So the answers are $(a),(b)$ and $(d)$.