I was going through the text "Discrete Mathematics and its Application" by Kenneth Rosen (5th Edition) where I am across the definition of equivalence relation and felt that it is one sided.
Definition: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive.
Now let us analyze the situation of what equivalence is meant to us intuitively.
Let there be a binary relation $R$ defined on a set $A$. Now we suppose that $R$ be reflexive, symmetric and transitive.
So we have for $a,b,c \in A$
$a R a$ (by the reflexive property of R)
if $a R b$ then $b R a$ (by the symmetric property of R)
if $a R b$ and $b R c$ then $aRc$ (by the transitive property of R)
Intuitively we can satisfy ourselves with the fact that the above are the necessary conditions for $R$ to be equivalent. So "if $R$ is reflexive, symmetric and transitive, then $R$ is an equivalence relation"
Now working our intuition for equivalence relation $\sim$ we note the following.
Let $\sim$ be an equivalence relation on a set A, then for $a,b,c \in A$ we have,
$a \sim a$ (by the intuitive knowledge of what $\sim$ means)
if $a\sim b$ then $b \sim a$ (by the intuitive knowledge of what $\sim$ means)
if $a\sim b$ and $b \sim c$ then $a\sim c$ (by the intuitive knowledge of what $\sim$ means)
Now we see that (1) implies $\sim$ is reflexive, (2) implies that $\sim$ is symmetric and (3) implies that $\sim$ is transitive.
So we have "if $\sim$ is an equivalent relation then $\sim$ is reflexive, symmetric and transitive"
From the two intuitive implications we can conclude that A relation on a set A is called an equivalence relation if and only if it is reflexive, symmetric, and transitive. and not what the book says. This definition makes quite sense unlike the book definition which says that if $R$ fails to be either reflexive or symmetric or transitive then $R$ may or may not be an equivalence relation, which after all gives a weird feeling.
Correct me if my logic is wrong.
