How does the cartesian product of set A satisfy an equivalence relation?

Question

Suppose A={a,b,c,d}. Why is A×A an equivalence relation?

Specifically, how would one show the Property of Transitivity simply?

Is it possible to show aRb, bRc ⟹ aRc in this way?

Generally speaking, how is the Cartesian Product of a set like A an equivalence relation?

Note: The professor does not want me to show x ∈ ℝ and the consequence, x − x = 0 ∈ ℤ. He does not want a rigorous proof or anything of the like. Just the intuition.

Answer 1

It's all about the notations and rigorous definitions.

What you write as $a\,R\,b$ is really defined as $R\subseteq A\times A$ and $\bf{(a,b)\in R}$.

$A\times A$ is the relation where every element is related to every element.

Answer 2

Let $ A=\{a,b,c,d\}$. $A\times A$ simply has every order pair generated from the elements. So you can prove the Property of Transitivity: Take $a,b,c\in A\times A$. Suppose $(a,b)\in A\times A$ and $(b,c)\in A\times A$. Immediately you have $(a,c)\in A\times A$, because $A\times A$ includes every pair of elements.

Bold words may give you some "intuition". In general, any relation on $A$ is a subset of $A\times A$.

Answer 3

The cartesian product contains $(x,y)$ for all $(x,y)\subset A$. So $aRc$ is always true. Logically speaking, this means $aRb\land bRc\Rightarrow aRc $, but you don't need to look at $aRb$ or $bRc$ to know that $aRc$ is true.