I am reading through Devlin's "Joy of sets".
I am up to the section on relations, and I wanted to check I had understood what a relation is and how it is defined in relation to a set.
Devlin states that a if $P$ is a property that applies to pairs of elements of a set $x$, then it is often called a binary relation ($R$ hereafter) on $P$ though strictly speaking the relation concerns the set:
{$(a,b)|a \in X \land b \in X \in P(x,y)$}
This suggests to me that $R$ is a set, and it is a set of elements of $X$ which make $P$ true. Is this correct?
For example: $X =$ {1, 2, 3, 4, 5}
If $R_1 = 2a$ where $a \in X$
Then $R_1 = $ {2, 4}
Because a unary relation on $X$ is defined to be a subset of $X$. Which means, although multiplying every element of $X$ by 2 gives {2, 4, 6, 8, 10}, only $2, 4 \in X$
If $R_2 = 2a + b | 2a + b$ is even
Then $R_2$ = {(1,2), (1,4), (2,2), (2,4), (3,2), (3,4), (4,2), (4,4), (5,2), (5,4)}
Because the pairs in $R_2$ are all of the ordered pairs of $X^2$ that make $R_2$ true.
Final example, if $R_3 = a + b + c | a + b + c = 3k, k \in$ Z
Then $R_3 = $ {(1,1,1), (1,1,4),...,(5,5,2), (5,5,5)}
Because the triples are all elements of $X^3$ which make $R_3$ true.
I ask because I am was struggling to appreciate the difference between $R$ and an equivalence relation - but that seems to be a relationship which is also reflexive, symmetric, and transitive.
Thank you in advance for any help or guidance.
Yes, a binary relation on $X$ is a subset of $X \times X$, and so it is also a set.
And yes, an equivalence relation is a special type of binary relation which is, as you said, reflexive, symmetric, and transitive.