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).