Need help with this Foundations of Mathematics homework problem. I received feedback from my professor for this problem and he said Quotient set description needs considerably more detail. Hint: Can you relate the quotient set to a well-known number system? Below is my work for this problem. Make sure handwriting is neat and readable.

1. K = ? x (? ? {0}) is the provided set which a relation ? is determined by (a,b) ? (c,d) iff ad = bc.

We will prove that ? is an equivalence relation. If you want to prove that ? is an equivalence relation, we are required to prove that ? is reflexive , symmetric, and transitive.

(i) Reflexive: Let (a,b)?K, we are required to prove that (a,b) ? (a,b).

Now since the commutative under multiplication is ?, we will be able to write

ab = ba (where a,b ? ??{0})

? (a,b) ? (b,a)

For this reason, this proves that ? is reflexive.

(ii) Symmetric: Let (a,b) ? (c,d), where (a,b) and (c,d) are elements of K, we are required to prove that (c,d) ? (a,b).

Now, (a,b) ? (c,d)

? ad = bc

With Z being the commutative under multiplication, we can therefore write it as

da = cb (since ad = da and bc = cb) or cb = da

? (c,d) ? (a,b)

For this reason, this proves that ? is symmetric.

(iii) Transitive: Let (a,b), (c,d) and (e,f) be in K such that

(a,b) ? (c,d) and (c,d) ? (e,f)

Then we are required to prove that

(a,b) ? (e,f)

Now we have

(a,b) ? (c,d) ? ad = bc

and (c,d) ? (e,f) ? cf = de

Here, ef = de gives us c = (As f ? 0)

Using this in

ad = bc

? ad = b(

? a = (cancel d on both sides and note that d ? 0)

? af = be

? (a,b) ? (f,e)

Earlier in the problem, ? is reflexive, which allows us to now write

(a,b) ? (f,e) ? (e,f)

that is, (a,b) ? (e,f)

? ? is transitive.

For this reason, it follows that ? is an equivalence relation.

Now we are required to explain the subsequent quotient set. Relation ? that is on set K is the quotient set, which is described as the set of all equivalence classes of relation ?.

Ex: x = (a,b)?K

We define

[x] = [(a,b)] = {y?K ? y?K}

= {y = (c,d)?K ? (c,d) ? (a,b)}

= {y = (c,d)?K ? (a,b) ? (c,d)}

= {y = (c,d)?K ? ad = bc}

Then the quotient set is defined as

K?? = {[x] ? x?K}