### Answer to the Statement

The statement "All relations are functions" is **false**.

### Explanation

1. **Definition of Relation**: A relation is any set of ordered pairs (x, y). It can contain multiple pairs with the same first element (x-value) and different second elements (y-values).

2. **Definition of Function**: A function is a specific type of relation where each input (x) is associated with exactly one output (y). This means that no two pairs in a function can have the same first element with different second elements.

3. **Counterexample**: For instance, the relation {(1, 2), (1, 3)} is not a function because the input 1 corresponds to two different outputs (2 and 3). However, the relation {(1, 2), (2, 3)} is a function because each input has a unique output.

Thus, while all functions are relations, not all relations qualify as functions.

Question

### Answer to the Statement

The statement "All relations are functions" is **false**.

### Explanation

1. **Definition of Relation**: A relation is any set of ordered pairs (x, y). It can contain multiple pairs with the same first element (x-value) and different second elements (y-values).

2. **Definition of Function**: A function is a specific type of relation where each input (x) is associated with exactly one output (y). This means that no two pairs in a function can have the same first element with different second elements.

3. **Counterexample**: For instance, the relation {(1, 2), (1, 3)} is not a function because the input 1 corresponds to two different outputs (2 and 3). However, the relation {(1, 2), (2, 3)} is a function because each input has a unique output.

Thus, while all functions are relations, not all relations qualify as functions.

Knowee AI · Accepted Answer

### Answer to the Statement

The statement "All relations are functions" is **false**.

### Explanation

1. **Definition of Relation**: A relation is any set of ordered pairs (x, y). It can contain multiple pairs with the same first element (x-value) and different second elements (y-values).

2. **Definition of Function**: A function is a specific type of relation where each input (x) is associated with exactly one output (y). This means that no two pairs in a function can have the same first element with different second elements.

3. **Counterexample**: For instance, the relation {(1, 2), (1, 3)} is not a function because the input 1 corresponds to two different outputs (2 and 3). However, the relation {(1, 2), (2, 3)} is a function because each input has a unique output.

Thus, while all functions are relations, not all relations qualify as functions.

Decide if the following statement about functions is true or false:All relations are functions.

Question

Decide if the following statement about functions is true or false:

Solution

Answer to the Statement

Explanation

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