Let `A={a,b,c,d}` and `B={x,y,z}`. Which of the following are relations from A to B? `{(a,x),(b,y),(c,z),z}`

Answer

To determine which of the given sets are relations from set A to set B, we need to understand what a relation is. A relation from set A to set B is defined as a subset of the Cartesian product A × B, which consists of ordered pairs (a, b) where 'a' is an element from set A and 'b' is an element from set B.

Given:
- Set A = {a, b, c, d}
- Set B = {x, y, z}

The relation we need to evaluate is: {(a, x), (b, y), (c, z), z}.

### Step-by-step Solution:

1. **Identify the elements of the relation:**
The given relation is {(a, x), (b, y), (c, z), z}. Here, we have three ordered pairs: (a, x), (b, y), and (c, z), and an additional element z.

2. **Check if each element is an ordered pair:**
- (a, x) is an ordered pair where 'a' is from set A and 'x' is from set B.
- (b, y) is an ordered pair where 'b' is from set A and 'y' is from set B.
- (c, z) is an ordered pair where 'c' is from set A and 'z' is from set B.
- z is not an ordered pair; it is just a single element from set B.

3. **Determine if the set is a relation:**
A relation must consist only of ordered pairs. Since z is not an ordered pair, it cannot be included in a relation from A to B.

4. **Conclusion:**
Since the element z is not an ordered pair, the set {(a, x), (b, y), (c, z), z} is **not** a relation from A to B.

### Final Answer:
The set {(a, x), (b, y), (c, z), z} is not a relation from A to B.