Let= (A,BC) and Y= (1,2,3,4). Which of the following relations A,B, and C defined below are functions from X to Y?

A= {(A,1),(B,2)( C,3)}
B= for all (x,y) E X X Y {X.Y) E B means that X is a vowel and Y is even
C= C define by the arrow diagram

To determine which of the relations \( A \), \( B \), and \( C \) are functions from \( X \) to \( Y \), we need to verify whether each relation satisfies the definition of a function. A function from \( X \) to \( Y \) is a relation where each element in the domain \( X \) is paired with exactly one element in the codomain \( Y \).

### Definitions:
- Let \( X = \{A, B, C\} \).
- Let \( Y = \{1, 2, 3, 4\} \).

### Let's examine each relation:

#### Relation \( A \):
\( A = \{(A, 1), (B, 2), (C, 3)\} \).

- In this relation, every element of \( X = \{A, B, C\} \) is mapped to exactly one element in \( Y = \{1, 2, 3, 4\} \).
- No element in \( X \) is mapped to more than one element in \( Y \).

**Conclusion:** \( A \) is a valid function because each element in \( X \) is paired with exactly one element in \( Y \).

#### Relation \( B \):
This relation is defined as:
- For all \( (x, y) \in X \times Y \), \( (x, y) \in B \) if and only if \( x \) is a vowel and \( y \) is an even number.

Let's break it down:
- In the domain \( X = \{A, B, C\} \), none of the elements \( A \), \( B \), or \( C \) are vowels (they are letters, but not vowels).
- Since \( B \) requires \( x \) to be a vowel, this means no element in \( X \) satisfies the condition.

**Conclusion:** \( B \) does not define a valid function from \( X \) to \( Y \) because there are no vowels in \( X \). As a result, no valid pairing can occur.

#### Relation \( C \):
\( C \) is defined by the arrow diagram (not provided here). However, for a relation to be a function:
- Every element of \( X \) must have exactly one arrow pointing to some element in \( Y \).

If you provide more details on the arrow diagram, I can verify whether \( C \) is a function or not. Based on this incomplete information, I cannot determine the status of \( C \).

### Summary:
- \( A \) is a function.
- \( B \) is **not** a function.
- \( C \) requires more information (the arrow diagram).

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Would you like more details or have any questions?

Here are 5 related questions to expand this topic:
1. What are the properties that define a mathematical function?
2. Can a relation be a function if multiple elements from the domain map to the same element in the codomain?
3. How does the concept of domain and codomain apply to real-world scenarios?
4. What is the significance of the vertical line test for determining if a graph represents a function?
5. How can we represent a function using a set of ordered pairs, a table, and a graph?

**Tip:** When checking if a relation is a function, ensure each input (element from the domain) has only one output (element from the codomain).

Let X= (A,B,C) and Y= (1,2,3,4). Which of the following relations A, B, and C defined below are functions from X to Y?
A= {(A,1),(B,2),(C,3)}
B= for all (x,y) ∈ X×Y, (x,y) ∈ B means that x is a vowel and y is even
C= C defined by the arrow diagram

Explore whether the relations A = {(A,1),(B,2),(C,3)}, B (based on vowel and even pairing), and C (arrow diagram) are valid functions from X = {A,B,C} to Y = {1,2,3,4}. Learn about the properties of functions and how to identify them in sets. This problem is suited for students in Grades 9-12.

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