Anonymous
Anonymous asked in Science & MathematicsMathematics · 3 years ago

# Math question?

What is an equivalence relation on a set X? (ii) Consider the relations R

and S, defined on the set X = {1, 2, . . . , 99} as follows.

xRy ⇐⇒ x + y is a multiple of 9,

xSy ⇐⇒ x − y is a multiple of 9.

One of R and S is an equivalence relation, the other is not. Determine which is which and

justify your answers. (iii) For the equivalence relation from part (ii), into how many classes

does it partition the set X?

Relevance

(i) In general, a relation Q on a set X is a just a subset of X^2. Q being a relation, we conventionally write (a,b) ∈ Q as aQb. An equivalence relation E on a set X is a relation on X that has the following 3 properties:

1) reflexivity: for every element x ∈ X, we have xEx.

2) symmetry: if aEb then bEa.

3) transitivity: if aEb and bEc, then aEc.

(ii) Claim: S is an equivalence relation on X. Proof: take any x ∈ X. Then x - x = 0 = 9*0, so xSx showing that S is reflexive. Check. Next, suppose aSb. That means that a - b = 9k for some integer k. Then b - a = 9(-k). Since -k is also an integer, we have bSa, showing that S is symmetric. Check. Finally suppose aSb and bSc. Then

a - b = 9m and

b - c = 9n,

for some integers m and n. Adding the above 2 equations, we have

a - c = 9(m+n).

Since m+n is also an integer we have aSc, showing that S is transitive, q.e.d.

On the other hand R is not reflexive (e.g. 1+1 is not a multiple of 9, so (1,1) ∉ R), so (even though it is symmetric and transitive) R isn't an equivalence relation.

(iii) S partitions X into 9 equivalence classes, namely its residue classes mod 9.

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• Professional Internet troll posts text alleged to be a technical question about a mathematics topic it probably does not understand:

"Math question? What is an equivalence relation on a set X? (ii) Consider the relations R

and S, defined on the set X = {1, 2, . . . , 99} as follows.

xRy ⇐⇒ x + y is a multiple of 9,

xSy ⇐⇒ x − y is a multiple of 9.

One of R and S is an equivalence relation, the other is not. Determine which is which and

justify your answers. (iii) For the equivalence relation from part (ii), into how many classes

does it partition the set X?"

Likewise for "define your operators and notation".

The troll in question probably copied the text from some book it never read.

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