let A and B be 2 sets. prove that if A is subset of set B, then A intersect complementB is empty

We are tasked with proving the following statement:

**Statement**: If \( A \subseteq B \), then \( A \cap B^c = \emptyset \).

Here, \( A \subseteq B \) means that every element of set \( A \) is also in set \( B \), and \( B^c \) represents the complement of set \( B \), i.e., the set of elements that are not in \( B \).

### Proof:

1. **Assumption**:  
   \( A \subseteq B \).  
   This means that if \( x \in A \), then \( x \in B \). In other words, every element of \( A \) is contained in \( B \).

2. **Complement of \( B \)**:  
   The complement of set \( B \), denoted by \( B^c \), is defined as:  
   \[
   B^c = \{x \mid x \notin B\}
   \]
   So, any element of \( B^c \) is not in \( B \).

3. **Intersection of \( A \) and \( B^c \)**:  
   We are tasked with proving that the intersection \( A \cap B^c = \emptyset \).  
   That is, we need to show that there is no element \( x \) that belongs to both \( A \) and \( B^c \).

4. **Suppose there is an element in the intersection**:  
   Let \( x \in A \cap B^c \).  
   By the definition of intersection, \( x \in A \) and \( x \in B^c \).

   - Since \( x \in A \) and \( A \subseteq B \), this implies that \( x \in B \).
   - But \( x \in B^c \) means \( x \notin B \).

   This is a contradiction because \( x \) cannot be both in \( B \) and not in \( B \) at the same time.

5. **Conclusion**:  
   Therefore, our assumption that such an \( x \) exists is false.  
   Hence, \( A \cap B^c = \emptyset \), which means that there are no elements that are in both \( A \) and the complement of \( B \).

Thus, we have proven that if \( A \subseteq B \), then \( A \cap B^c = \emptyset \).

---

Would you like any further clarification or details on this proof? Here are some related questions you might want to consider:

1. What is the complement of a set in a universal set?
2. Can a set have a non-empty intersection with its own complement?
3. How would the proof change if \( A \) is not a subset of \( B \)?
4. What is the difference between a strict subset and a subset?
5. How does set theory relate to logic in terms of proving properties?

**Tip**: Remember that a set and its complement are always disjoint!

Let A and B be two sets. Prove that if A is a subset of set B, then A intersect complement of B is empty.

Explore the proof that if set A is a subset of set B, then A intersect complement of B is empty. This problem covers key set theory concepts such as subset, complement, and intersection. Suitable for high school students studying set theory.

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