Proof of equivalence relations

Question

I have just started my math class and I think I might not be completely understanding the 3 properties of a equivalence relation: reflexivity, symmetry, transitivity.

I have these 2 examples and I would like to know if my steps are good or what I am missing.

What would be the main difference by the bigger or equal to 0 and bigger than 0?

Knowing $a, b \in \mathbb{Z}$:

A. $a \sim b$ if $ab \geq 0$

for reflexivity: $a\sim a$ if $a^2 \geq 0$ then yes

for symmetry: $a \sim b$ if $b \sim a$ then $ab \geq 0$ if $ba \geq 0$ then yes

for transitivity: $a \sim b$ if $b \sim a$ then $a \sim c$ then $ab \geq 0$ if $bc \geq 0$ if $ac \geq 0$ then no

B. $a \sim b$ if $ab > 0$

for reflexivity: $a \sim a$ if $a-a=0$ so yes

for symmetry: $a \sim b$ if $b \sim a$ then $a - b = c \in \mathbb{Z}$ so $b - a = -c \in \mathbb{Z}$ then yes

for transitivity: $a \sim b$ if $b \sim a$ then $a \sim c$ Here I have no idea how to prove it

Thank you for your feedback!

Answer 1

Let's go through the second carefully, then you can do the first yourself. Define $a \sim b$ if and only if $ab>0$.

• Reflexivity means $a \sim a$ for all $a \in \mathbb{Z}$. Let $a \in \mathbb{Z}$, then $a \sim a \iff a^2 > 0$ which is only true if $a\ne 0$, hence the relation is not reflexive
• Symmetry means $a\sim b \iff b \sim a$ for all $a,b \in \mathbb{Z}$. Let $a,b \in \mathbb{Z}$ then $a\sim b \iff ab > 0 \iff ba > 0 \iff b \sim a$, so indeed the relation is symmetric
• Transitivity means $a\sim b, b\sim c \iff a \sim c$ for all $a,b,c \in \mathbb{Z}$. Let $a,b,c \in \mathbb{Z}$ then $$ \begin{split} a\sim b,b\sim c &\iff ab>0,bc>0\\ &\iff a,b \text{ have same sign and } b,c \text{ have same sign}\\ &\iff a,b,c \text{ all have the same sign}\\ &\iff ac > 0 \\ &\iff a \sim c, \end{split} $$ so the relation is transitive