We are tasked with showing that for an injective mapping $f: X \to Y$ and subsets $A, B \subseteq X$, the following holds:

$f(A \cap B) = f(A) \cap f(B)$

Step 1: Show $f(A \cap B) \subseteq f(A) \cap f(B)$

We will start by showing that $f(A \cap B) \subseteq f(A) \cap f(B)$.

• Take any element $y \in f(A \cap B)$. By definition, this means there exists some $x \in A \cap B$ such that $f(x) = y$.
• Since $x \in A \cap B$, it follows that $x \in A$ and $x \in B$.
• Therefore, $f(x) \in f(A)$ and $f(x) \in f(B)$, which means $y \in f(A) \cap f(B)$.

Thus, every element of $f(A \cap B)$ is also in $f(A) \cap f(B)$, so: $f(A \cap B) \subseteq f(A) \cap f(B)$

Step 2: Show $f(A) \cap f(B) \subseteq f(A \cap B)$

Now, we will show the reverse inclusion $f(A) \cap f(B) \subseteq f(A \cap B)$.

• Take any element $y \in f(A) \cap f(B)$. By definition, this means there exist elements $x_1 \in A$ and $x_2 \in B$ such that: $f(x_1) = y \quad \text{and} \quad f(x_2) = y$
• Since $f$ is injective, it follows that $x_1 = x_2$, because injectivity implies that $f(x_1) = f(x_2)$ only if $x_1 = x_2$.
• Therefore, there exists some $x \in A \cap B$ such that $f(x) = y$.

Thus, every element of $f(A) \cap f(B)$ is also in $f(A \cap B)$, so: $f(A) \cap f(B) \subseteq f(A \cap B)$

Conclusion

Since we have shown both $f(A \cap B) \subseteq f(A) \cap f(B)$ and $f(A) \cap f(B) \subseteq f(A \cap B)$, it follows that: $f(A \cap B) = f(A) \cap f(B)$

This completes the proof.

Do you have any questions or would you like further details?

5 Related Questions:

1. How does this proof change if the function $f$ is not injective?
2. What is the significance of injectivity in set theory and mappings?
3. Can you provide an example where $f(A \cap B) \neq f(A) \cap f(B)$ for a non-injective function?
4. What happens if $A$ or $B$ is the empty set in this scenario?
5. How can this result be extended to other set operations, like union or difference?

Tip:

Injective functions (one-to-one) preserve distinctness, making them useful when proving equalities involving set images under mappings.