Question

If \( X = \{ a, b, c \} \) and \( Y = \{ 1, 2, 3 \} \) and the mapping f is given by \( f(a)=2, f(b)=3, f(c)=1 \), then

• A. \( f(X) \subset Y \)
• B. \( f(X) = Y \)
• C. \( f(X) \supset Y \)
• D. All of these

Hint:

Remember the set theory definitions: - \( A \subseteq B \) (A is a subset of B) if every element of A is in B. - \( A = B \) if \( A \subseteq B \) and \( B \subseteq A \). By this definition, if \( A = B \), then \( A \subseteq B \) and \( A \supseteq B \) are both true.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to find the range of the function f, which is denoted by \( f(X) \), and compare it with the codomain Y. The range is the set of all output values of the function.
Step 2: Detailed Explanation:
The domain is \( X = \{a, b, c\} \).
The codomain is \( Y = \{1, 2, 3\} \).
The mapping is given as: \[ f(a) = 2 \] \[ f(b) = 3 \] \[ f(c) = 1 \] The range of the function, \( f(X) \), is the set of all images of the elements of X.
\[ f(X) = \{f(a), f(b), f(c)\} = \{2, 3, 1\} \] Reordering the elements for clarity, we get: \[ f(X) = \{1, 2, 3\} \] Now, we compare \( f(X) \) with \( Y \): \[ f(X) = \{1, 2, 3\} \quad \text{and} \quad Y = \{1, 2, 3\} \] We can see that \( f(X) = Y \).
Step 3: Analyzing the Options:
Based on the fact that \( f(X) = Y \):
(i) \( f(X) \subset Y \): This means "f(X) is a subset of Y". Since every element of f(X) is also in Y, this statement is true. (Note: A set is a subset of itself).
(ii) \( f(X) = Y \): This means "f(X) is equal to Y". As we found, this is true.
(iii) \( f(X) \supset Y \): This means "f(X) is a superset of Y". Since every element of Y is also in f(X), this statement is true.
Since statements (i), (ii), and (iii) are all correct, the correct option is (iv) All of these.
Step 4: Final Answer:
The range of the function is equal to its codomain, which makes all three relationships true. Therefore, the correct option is (iv).