Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w}; `S={(m/n , p/q)"m , n , pandqa r ei n t e g e r ss u c ht h a tn ,q"!="0andq m = p n"}` . Then (1) neither R nor S is an equivalence relation (2) S is an equivalence relation but R is not an equivalence relation (3) R and S both are equivalence relations (4) R is an equivalence relation but S is not an equivalence relation

Options

A. R and S both are equivalence relations.
B. R is an equivalence relation but S is not an equivalence relation.
C. Neither R nor S is an equivalence relation.
D. S is an equivalence relation but R is not an equivalence relation.

Explanation

To determine whether the relations R and S are equivalence relations, we need to check the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

### Step 1: Analyze Relation R
**Definition of R:**
R = {(x, y) | x, y are real numbers and x = wy for some rational number w}.

**Check Reflexivity:**
For reflexivity, we need to check if (x, x) ∈ R for all x.
- Choose any real number x.
- We can set w = 1 (which is a rational number), thus x = 1 * x.
- Therefore, (x, x) ∈ R for all x.
- **Conclusion:** R is reflexive.

**Check Symmetry:**
For symmetry, we need to check if (x, y) ∈ R implies (y, x) ∈ R.
- Suppose (x, y) ∈ R, meaning x = wy for some rational w.
- Rearranging gives y = (1/w)x.
- Since w is rational, 1/w is also rational (as long as w ≠ 0).
- Thus, (y, x) ∈ R.
- **Conclusion:** R is symmetric.

**Check Transitivity:**
For transitivity, we need to check if (x, y) ∈ R and (y, z) ∈ R implies (x, z) ∈ R.
- Suppose (x, y) ∈ R and (y, z) ∈ R.
- This means x = wy for some rational w and y = vz for some rational v.
- Substituting gives x = w(vz) = (wv)z.
- Since the product of two rational numbers (w and v) is rational, (x, z) ∈ R.
- **Conclusion:** R is transitive.

### Step 2: Analyze Relation S
**Definition of S:**
S = {(m/n, p/q) | m, n, p, q are integers such that n ≠ 0 and qm = pn}.

**Check Reflexivity:**
For reflexivity, we need to check if (m/n, m/n) ∈ S for all m/n.
- Here, qm = pn becomes q(m) = p(m) when m = p.
- This holds true since we can choose q = n and p = n.
- **Conclusion:** S is reflexive.

**Check Symmetry:**
For symmetry, we need to check if (m/n, p/q) ∈ S implies (p/q, m/n) ∈ S.
- If (m/n, p/q) ∈ S, then qm = pn.
- Rearranging gives qp = nm, which is the same as (p/q, m/n) ∈ S.
- **Conclusion:** S is symmetric.

**Check Transitivity:**
For transitivity, we need to check if (m/n, p/q) ∈ S and (p/q, r/s) ∈ S implies (m/n, r/s) ∈ S.
- From (m/n, p/q) ∈ S, we have qm = pn.
- From (p/q, r/s) ∈ S, we have qp = rs.
- We can substitute to show that qm = rs, thus (m/n, r/s) ∈ S.
- **Conclusion:** S is transitive.

### Final Conclusion
Both R and S satisfy reflexivity, symmetry, and transitivity. Therefore, both R and S are equivalence relations.

### Answer
(3) R and S both are equivalence relations.