Let A be a finite set of size k and R a relation on the...

Jayvion Caldwell

Jayvion Caldwell

Answered

2022-07-17

Let A be a finite set of size k and R a relation on the power set P(A) defined by R = { ( A , B ) : | A | = | B | }
1. Show that A is an equivalence relation.
2. Let a A. What is the size of the equivalence class of {a}?
3. Let a,b be two different elements of A. What is the size of the equivalence class of {a,b}?

Answer & Explanation

grocbyntza

grocbyntza

Expert

2022-07-18Added 25 answers

Step 1
There are three criteria that must be followed in order to prove R is an equivalence relation,
1. Reflexive
2. Symmetric
3. Transitive
Step 2
To get you started, with generic sets B , C , D P ( A ), see that we always have | B | = | B |
so R is reflexive. This would mean ( B , B ) R for some B A
To show it's symmetric, it requires you to show that if | B | = | C | , then | C | = | B | .
To show it's transitive, it requires you to show if | B | = | C | , and | C | = | D | , then | B | = | D |
Nash Frank

Nash Frank

Expert

2022-07-19Added 10 answers

Step 1
Informally, under this equivalence relation two subsets are equivalent when they have the same size.
Thus, the equivalence class of {a} consists of all subsets of A with cardinality/size equal to one.
Thus the size of this equivalence class is k = | A | .
Step 2
The equivalence class of {a,b} consists of all two element subsets of A. Thus the size of this equivalence class is ( k 2 ) = k ( k 1 ) 2

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