How many reflexive relations on set A with 4 elements?

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In summary, there are 4 reflexive relations on set A if A has 4 elements. This is because a reflexive relation on a set is any subset of the set's Cartesian product, and for a set with 4 elements, there are 4 specific pairs that must be included in a reflexive relation. Therefore, there are $2^{4}= 16$ subsets and out of those, $2^{12}= 4096$ subsets are reflexive relations.
  • #1
lemonthree
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Question: How many reflexive relations are on set A if A has 4 elements?

I'm thinking that the answer to this question is 4, but I don't know whether it is truly that simple.
I know that a reflexive relation, R, occurs when for all x in A, x R x.
so let's say A = {1,2,3,4}.
1~1
2~2
3~3
4~4.
So there are 4 "relations" here. I believe the answer to this is 4, *or* it could be 1 as A~A. Is it right?
 
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  • #2
First, do you know what a "relation" is? You seem to be completely misunderstanding this problem.

A relation on set A is any subset of AxA. In this case, since A has 4 members, say A= {a, b, c, d}, AxA has 16 members, AxA= {(a, a), (a, b), (a, c), (a, d), (b, a), (b, b), (b, c), (b, d), (c, a), (c, b), (c, c), (c, d), (d, a), (d, b), (d, c), (d, d)}. Every such pair may or may not be in a subset of AxA. Since A has 16 pairs there are 16 "yes" or "no" choices to determine whether or not a specific pair is in AxA or not. There are $2^{16}= 65536$ subsets. (In general, if a set contains n members then it has $2^n$ subsets.)

If the only condition is that the relation be reflexive then we are only requiring that the subset contain (a, a), (b, b), (c, c), and (d, d). That leaves 12 other pairs that might of might not be in the set so there are $2^{12}= 4096$ such sets.
 
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  • #3
Thank you for helping to clear my misunderstanding! It seems to remind me of the power set a little.

Just to make sure I'm understanding it right, I can say that all the 4096 reflexive relations on set A *will* definitely contain the 4 sets (a, a), (b, b), (c, c), and (d, d)?
 
  • #4
Yes, that's what "reflexive" means!
 

Related to How many reflexive relations on set A with 4 elements?

1. What is a reflexive relation?

A reflexive relation is a type of binary relation in mathematics and logic where every element is related to itself. In other words, for any element a in a set A, the relation R is reflexive if (a, a) ∈ R. This means that every element is related to itself in the given relation.

2. How is a reflexive relation different from a symmetric or transitive relation?

A reflexive relation is different from a symmetric or transitive relation because it only requires that every element is related to itself, while symmetric and transitive relations have additional requirements. A symmetric relation requires that if (a, b) ∈ R, then (b, a) ∈ R, and a transitive relation requires that if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

3. What are some examples of reflexive relations?

Some examples of reflexive relations include the equality relation (where every element is related to itself), the "is a subset of" relation (where every set is a subset of itself), and the "is a parent of" relation (where every person is their own parent).

4. Can a relation be reflexive and symmetric at the same time?

Yes, a relation can be both reflexive and symmetric. For example, the "is equal to" relation is reflexive because every element is equal to itself, and it is also symmetric because if a = b, then b = a.

5. Why is it important to define reflexive relations?

Defining reflexive relations is important because they have many applications in mathematics, computer science, and other fields. They are used to define important concepts such as equivalence relations and partial orders, and they are also used in algorithms and data structures. By understanding and defining reflexive relations, we can better understand and solve problems in various areas of study.

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