Proving that $(a_1, b_1)\sim(a_2, b_2)\Leftrightarrow\ a_1=a_2\land b_1=b_2$

Question

This assingment is preparation for exam.

I need to prove with $(a_1, b_1)\sim(a_2, b_2)\Leftrightarrow\ a_1=a_2\land b_1=b_2$ that $\sim$ is equivalence relatio.

Can you tell me how to do this.

Thanks!!!

Answer 1

Simple. There are three properties an equivalence relation above the set $X$ must fulfill:

• For any $x\in X$, you must have $x\sim x$ (reflexive)
• For any $x,y\in X$ for which $x\sim y$, you must have $y\sim x$ (symmetric)
• For any $x,y,z\in X$, for which $x\sim y$ and $y\sim z$, you must have $x\sim z$ (transitive)

Try proving these properties one by one for your relation. Post results, then we will help you if something goes wrong.

Note: i intentionaly wrote $x$ instead of $(a,b)$. In your case, the set $X$ contains ordered pairs, which may be confuzing but does not change a thing.