5. For each of the following, prove that the relation is an equivalence relation

Question

                    5. For each of the following, prove that the relation is an equivalence relation. Then give information about the equivalence class specified.                 

                    b) The relation R on N (natural number) given by mRn iff m and n have the same digit in the tens places. Find an element of 106/R that is less than 50; between                     150 and 300; greater than 1,000. Find three such elements in the equivalence class 635/R.                 

                    f) For the set X={m,n,p,q,r,s}, let R be the relation on P(x) (powerset) given by A R B iff A and B have the same number of elements. List all the elements in                     [m]/R; in {m,n,p,q,r,s}/R. How many elements are in X/R? How many elements are in P(x)/R?                 

Explanation / Answer

B)

The relation R on natural numbers given by m R n if and only if m and n have the same digit in the tens places. Name an element of 106/R that is less than 50; between 150 and 300; greater than 1,000. Find three such elements in the equivalence class of 635/R

again you may easily check this is an equivalence relation..
Since 106 has 0 at tens, you may say there's no elements from 106/R less then 50, or you may name any one-digit number (i dunno what is meant here). 150< 200 <300, 1001>1000 seem to satisfy the conditions..
for 635 numbers 35, 235 and 1035 suit (if i understood the terms correctly)

F)

the number of elements in {m} is 1, namely, m. so A is in the equivalence class of {m} iff A has one element, that is:

{m}/R = {{m},{n},{p},{q},{r},{s}}

in general, the size of X/R is the number of elements of P(X) of cardinality |X| = k, which is:

720/(k!(6-k)!) or "6 choose k".

the number of elements in X/R are the number of elements of P(X) that have 6 members,

which is only 1, namely X itself.

there are a total of 7 elements in the set of equivalence classes, one for each possibile |X|:

{ }/R, the equivalence class of the empty set, all sets of P(X) with 0 elements, and:

sets of P(X) with 1 element ( = {m}/R),
sets of P(X) with 2 elements (= {m,n}/R)
sets of P(X) with 3 elements (= {m,n,p}/R)
sets of P(X) with 4 elements ( = {m,n,p,q}/R)
sets of P(X) with 5 elements ( = {m,n,p,q,r}/R)
and X itself ( = X/R).

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