Let f:A→Bbe a bijective function from set A to set B and let be some third set for which C ⊆ A
. Prove that: f(C)∩f(A−C)=∅

To prove that \( f(C) \cap f(A - C) = \emptyset \), where \( f: A \to B \) is a bijective function and \( C \subseteq A \), we proceed step by step.

### Proof:

#### 1. Understanding the sets:

- \( C \subseteq A \) means that \( C \) is a subset of \( A \).
- \( A - C \) is the complement of \( C \) with respect to \( A \). This set consists of all elements of \( A \) that are not in \( C \).
- \( f(C) \) represents the image of \( C \) under the function \( f \).
- \( f(A - C) \) represents the image of \( A - C \) under the function \( f \).

We are asked to prove that the intersection of \( f(C) \) and \( f(A - C) \) is empty, meaning that no element is common between the two sets.

#### 2. Assume the opposite:

Suppose \( f(C) \cap f(A - C) \neq \emptyset \). This means there exists some element \( y \in B \) such that:
\[
y \in f(C) \cap f(A - C)
\]
Thus, there exist elements \( x_1 \in C \) and \( x_2 \in A - C \) such that:
\[
f(x_1) = y \quad \text{and} \quad f(x_2) = y
\]
Since \( f \) is bijective (both injective and surjective), it is injective, meaning \( f(x_1) = f(x_2) \) implies that \( x_1 = x_2 \).

#### 3. Contradiction:

However, by definition, \( x_1 \in C \) and \( x_2 \in A - C \). This implies that \( x_1 \neq x_2 \), because \( x_1 \) is an element of \( C \) and \( x_2 \) is not. This contradicts the injectivity of \( f \).

#### 4. Conclusion:

Thus, our assumption that \( f(C) \cap f(A - C) \neq \emptyset \) must be false. Therefore, we conclude that:
\[
f(C) \cap f(A - C) = \emptyset
\]

This completes the proof.

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

### Related Questions:

1. What is the significance of a bijective function in set theory?
2. How does injectivity play a role in proving set intersections?
3. Can this proof be applied to non-bijective functions?
4. How would the proof change if \( f \) were not injective?
5. How do we visualize the relationship between \( f(C) \) and \( f(A - C) \)?

### Tip:
In set theory, bijective functions guarantee a one-to-one correspondence between sets, which is crucial in ensuring that images of disjoint sets remain disjoint.

Let f: A → B be a bijective function from set A to set B and let C be some third set for which C ⊆ A. Prove that: f(C) ∩ f(A − C) = ∅

Learn how to prove that the intersection of f(C) and f(A - C) is empty when f is a bijective function from set A to set B. This step-by-step proof involves set theory and the injectivity property of bijections. Suitable for advanced high school or undergraduate students.

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