I have a question about Equivalence relation.
There is a set A, and for this set I have a Equivalence relation C.
Then there is a subset B ⊂ A
My question is about this expression I don't understand: C ∩ (B x B )
How can be the Relation C intersected with the pair (BxB)? Is C a set? I don't understand how can a relation been intersected with a set?
Thank you
$C$ is a relation on $A$, which means that $C\subseteq A\times A$, i.e., it is a set of ordered pairs.
For example, if $A=\{0,1,2,3,4,5\}$, then $C=\{(0,0),(1,1),(2,2),(3,3),(4,4),(5,5),(0,1),(1,0),(2,3),(3,2)\}$ is an equivalence relation on $A$. (It corresponds to the partition $\{\{0,1\},\{2,3\},\{4\},\{5\}\}$.)
Now if $B\subseteq A$ then $B\times B$ is a set of ordered pairs of elements from $A$. (Namely, it contains all pairs where both coordinates are from $B$.) So it makes sense to intersect $C$ and $B\times B$.
For example, take $B=\{0,1,2\}$. Then $B\times B=\{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)\}$ and
$C\cap (B\times B) = \{(0,0),(1,1),(2,2),(0,1),(1,0)\}$. Notice that $C\cap (B\times B)$ is an equivalence relation on the set $B$.