https://file.aiquickdraw.com/mathbot/file/1727270280933tdjqfytc.jpg

The question asks which of the relations \(A\), \(B\), and \(C\) are functions from \(X\) to \(Y\). 

Let's analyze each relation:

### Given:
- \(X = \{a, b, c\}\)
- \(Y = \{1, 2, 3, 4\}\)

### Relation \(A\):
- \(A = \{(a, 1), (b, 2), (c, 3)\}\)
  
Relation \(A\) pairs each element from \(X\) to exactly one element in \(Y\). Therefore, \(A\) **is a function** because each element in \(X\) maps to one unique element in \(Y\).

### Relation \(B\):
- \(B\) is described as: for all \((x, y) \in X \times Y\), \((x, y) \in B\) means that \(x\) is a vowel, and \(y\) is even.

From the set \(X = \{a, b, c\}\), only \(a\) is a vowel. So, \(B\) would include pairs \((a, y)\) where \(y\) is even, i.e., \(B = \{(a, 2), (a, 4)\}\).

This is **not a function** because the element \(a\) in \(X\) maps to two elements (2 and 4) in \(Y\), violating the definition of a function (which requires each element in \(X\) to map to exactly one element in \(Y\)).

### Relation \(C\):
- \(C\) is defined by the arrow diagram.

From the diagram:
- \(a \to 1\)
- \(b \to 1\)
- \(c \to 3\)

This relation pairs each element in \(X\) to exactly one element in \(Y\), so \(C\) **is a function**.

### Conclusion:
- Relation \(A\) is a function.
- Relation \(B\) is not a function.
- Relation \(C\) is a function.

Would you like more details or explanations? Here are some related questions:

1. What are the characteristics of a function in set theory?
2. Can a function map to the same element in \(Y\) for different elements in \(X\)?
3. How do you identify whether a relation is a function from an arrow diagram?
4. What happens if a relation maps one element in \(X\) to multiple elements in \(Y\)?
5. Can a relation be a function if some elements of \(X\) are not mapped to any element in \(Y\)?

**Tip:** When checking if a relation is a function, always verify that each input from set \(X\) has exactly one output in set \(Y\).

Which of the relations A, B, C defined below are functions from X to Y?

Analyze the given relations A, B, and C defined from sets X = {a, b, c} and Y = {1, 2, 3, 4} to determine which are functions. This problem covers fundamental set theory concepts, including the definition of a function.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation