To determine whether the statement is true, let us carefully analyze the claim.

Problem Setup:

We are given a function $f: A \to B$ and subsets $X, Y \subseteq A$. The claim is:

$f(X) \cap f(Y) = f(X \cap Y)$

where:

• $f(X)$ represents the image of the subset $X$, i.e., $f(X) = \{f(x) \mid x \in X\}$, and similarly for $f(Y)$.
• $X \cap Y$ is the intersection of subsets $X$ and $Y$.

Step 1: Analyze $f(X) \cap f(Y)$

The intersection $f(X) \cap f(Y)$ contains all elements $b \in B$ such that $b = f(x_1)$ for some $x_1 \in X$ and $b = f(x_2)$ for some $x_2 \in Y$.

In other words: $f(X) \cap f(Y) = \{b \in B \mid b = f(x) \text{ for some } x \in X \cap Y \text{ or } x \in X \text{ and } x \in Y\}.$

Step 2: Analyze $f(X \cap Y)$

The image of $X \cap Y$ is: $f(X \cap Y) = \{f(x) \mid x \in X \cap Y\}.$

This directly represents the set of function values for elements that are in both $X$ and $Y$.

Step 3: Compare $f(X) \cap f(Y)$ and $f(X \cap Y)$

• Key Observation: It is possible that $f(x_1) = f(x_2)$ even though $x_1 \neq x_2$. This means that the function $f$ may not be injective (one-to-one). If $f$ is not injective, there may exist elements in $f(X) \cap f(Y)$ that do not correspond to any element in $X \cap Y$.

Example:

Let $A = \{1, 2\}$, $B = \{a\}$, and $f(1) = f(2) = a$. Take subsets $X = \{1\}$, $Y = \{2\}$:

• $X \cap Y = \emptyset$, so $f(X \cap Y) = f(\emptyset) = \emptyset$.
• $f(X) = \{a\}$, $f(Y) = \{a\}$, so $f(X) \cap f(Y) = \{a\}$.

Here, $f(X) \cap f(Y) \neq f(X \cap Y)$.

Conclusion:

The statement is false in general. It only holds if $f$ is an injective (one-to-one) function, where $f(x_1) = f(x_2) \implies x_1 = x_2$.

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Do you want a more detailed explanation or additional examples?
Here are five related questions to consider:

1. What are the conditions under which $f(X) \cap f(Y) = f(X \cap Y)$ is always true?
2. How does the surjectivity or injectivity of a function affect operations on subsets?
3. Can you provide an example where $f$ is surjective but the given statement fails?
4. What happens if $f$ is a bijection? Does the equality always hold?
5. How does the image of a function relate to other set operations like union or difference?

Tip: When comparing images of functions, always check whether the function is injective or surjective, as this determines many important properties.