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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

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$C$ is a subset of $A\times A$ –  ՃՃՃ Jan 17 '13 at 13:57
    
Thank you Psx. Ok, I understand, the subset are the pairs of elements tha –  Pedro Jan 17 '13 at 13:59
    
Thank you Psx. Ok, I understand, the subset are the pairs of elements that meet the relation. :-) –  Pedro Jan 17 '13 at 14:00
    
That's right. You're welcome. :-) –  ՃՃՃ Jan 17 '13 at 14:02

1 Answer 1

$C$ 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 $B$.

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Wonderfull, thank you. –  Pedro Jan 17 '13 at 14:31

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