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,x),(a,d)}`

Answer

To determine which of the given pairs are relations from set A to set B, we first need to understand what a relation from A to B is. A relation from set A to set B is a subset of the Cartesian product A × B, which consists of all possible 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}

### Step 1: Find the Cartesian Product A × B
The Cartesian product A × B consists of all possible ordered pairs formed by taking one element from A and one element from B.

A × B = {(a, x), (a, y), (a, z), (b, x), (b, y), (b, z), (c, x), (c, y), (c, z), (d, x), (d, y), (d, z)}

### Step 2: Identify the Given Relations
We need to check the following pairs to see if they belong to the relation from A to B:
1. (a, x)
2. (b, y)
3. (c, x)
4. (a, d)

### Step 3: Check Each Pair
1. **(a, x)**: This pair is valid because 'a' is in A and 'x' is in B.
2. **(b, y)**: This pair is valid because 'b' is in A and 'y' is in B.
3. **(c, x)**: This pair is valid because 'c' is in A and 'x' is in B.
4. **(a, d)**: This pair is **not valid** because 'd' is not in set B (it is in set A).

### Conclusion
The valid relations from A to B are:
- (a, x)
- (b, y)
- (c, x)

The pair (a, d) is **not** a relation from A to B.

### Final Answer
The pairs that are relations from A to B are:
- (a, x)
- (b, y)
- (c, x)

The pair (a, d) is **not** a relation from A to B.

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