If A = {x, y, z}, then the relation
`R={(x,x),(y,y),(z,z),(z,x),(z,y)}`, is

Options

A. symmetric
B. antisymmetric
C. transitive
D. both (a) and (b)

Answer

To determine the type of relation \( R \) defined on the set \( A = \{x, y, z\} \) with the relation \( R = \{(x,x), (y,y), (z,z), (z,x), (z,y)\} \), we will check if the relation is symmetric, anti-symmetric, and transitive.

### Step 1: Check for Symmetry
A relation is symmetric if for every pair \( (a, b) \) in \( R \), the pair \( (b, a) \) is also in \( R \).

- The pairs in \( R \) are:
- \( (x,x) \): symmetric (since \( (x,x) \) is the same)
- \( (y,y) \): symmetric (since \( (y,y) \) is the same)
- \( (z,z) \): symmetric (since \( (z,z) \) is the same)
- \( (z,x) \): not symmetric because \( (x,z) \) is not in \( R \)
- \( (z,y) \): not symmetric because \( (y,z) \) is not in \( R \)

Since \( (z,x) \) is in \( R \) but \( (x,z) \) is not, the relation is **not symmetric**.

### Step 2: Check for Anti-symmetry
A relation is anti-symmetric if for every pair \( (a, b) \) and \( (b, a) \) in \( R \), it must hold that \( a = b \).

- The pairs \( (x,x), (y,y), (z,z) \) are trivially anti-symmetric.
- For \( (z,x) \), we do not have \( (x,z) \) in \( R \), so it does not violate anti-symmetry.
- For \( (z,y) \), we do not have \( (y,z) \) in \( R \), so it does not violate anti-symmetry.

Since there are no pairs \( (a,b) \) and \( (b,a) \) where \( a \neq b \), the relation is **anti-symmetric**.

### Step 3: Check for Transitivity
A relation is transitive if whenever \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) must also be in \( R \).

- Check pairs:
- From \( (z,x) \) and \( (x,x) \), we should have \( (z,x) \) which is in \( R \).
- From \( (z,y) \) and \( (y,y) \), we should have \( (z,y) \) which is in \( R \).
- From \( (z,x) \) and \( (x,y) \) (not present), we do not need to check further.

Since all necessary conditions for transitivity are satisfied, the relation is **transitive**.

### Conclusion
The relation \( R \) is **anti-symmetric** and **transitive**, but **not symmetric**.