Identify the theorems and/or identities that justify each step in the derivation below.
If A and B are sets in a finite universe U, then
N(A ∩ B)	 = 		N 

(A ∩ B) ∩ U


---Select---
 = 		N 

U ∩ (A ∩ B)


---Select---
 = 		N 

U ∩ 

(A ∩ B)c
	c
 


---Select---
 = 		N 

U − (A ∩ B)c


---Select---
 = 		N(U) − N 

(A ∩ B)c

    	
---Select---
 = 		N(U) − N(Ac ∪ Bc)	    	
---Select---
 = 		N(U) − 

N(Ac) + N(Bc) − N(Ac ∩ Bc)


To identify the theorems and/or identities that justify each step, let's carefully analyze the steps and match them to standard set-theoretic principles:

---

1. **Step**:  
   \( N(A \cap B) = N((A \cap B) \cap U) \)  
   **Justification**: **Intersection with the universe**  
   The identity \( S = S \cap U \), where \( U \) is the universal set, holds because intersecting a set with \( U \) doesn't change the set.

---

2. **Step**:  
   \( N((A \cap B) \cap U) = N(U \cap (A \cap B)) \)  
   **Justification**: **Commutativity of intersection**  
   The order of intersection does not matter: \( S \cap T = T \cap S \).

---

3. **Step**:  
   \( N(U \cap (A \cap B)) = N(U \cap (A \cap B)^c)^c \)  
   **Justification**: **Complement rule**  
   The complement of a complement restores the original set: \( S = (S^c)^c \).

---

4. **Step**:  
   \( N(U \cap (A \cap B)^c)^c = N(U - (A \cap B)^c) \)  
   **Justification**: **Set difference as intersection with complement**  
   \( S - T = S \cap T^c \), which matches the transformation here.

---

5. **Step**:  
   \( N(U - (A \cap B)^c) = N(U) - N((A \cap B)^c) \)  
   **Justification**: **Principle of inclusion-exclusion for complements**  
   The number of elements in \( S - T \) equals the total elements in \( S \) minus the elements in \( T \).

---

6. **Step**:  
   \( N((A \cap B)^c) = N(A^c \cup B^c) \)  
   **Justification**: **De Morgan's laws**  
   \( (A \cap B)^c = A^c \cup B^c \), which relates complements of intersections to unions of complements.

---

7. **Step**:  
   \( N(A^c \cup B^c) = N(A^c) + N(B^c) - N(A^c \cap B^c) \)  
   **Justification**: **Principle of inclusion-exclusion for unions**  
   The cardinality of a union is the sum of individual cardinalities minus the overlap.

---

### Summary of Justifications:
1. **Intersection with the universe**  
2. **Commutativity of intersection**  
3. **Complement rule**  
4. **Set difference as intersection with complement**  
5. **Inclusion-exclusion for complements**  
6. **De Morgan's laws**  
7. **Inclusion-exclusion for unions**

---

Would you like me to elaborate on any of these justifications? Here are 5 related questions:

1. How do De Morgan's laws apply to set complements in real-world scenarios?  
2. Can you explain inclusion-exclusion principles with a practical example?  
3. How does the concept of the universal set simplify operations in set theory?  
4. What is the relationship between set differences and intersections with complements?  
5. Why is commutativity essential in set operations?  

**Tip**: When working with set theory, drawing Venn diagrams often clarifies the operations and helps visualize relationships.

Identify the theorems and/or identities that justify each step in the derivation below.
If A and B are sets in a finite universe U, then
N(A ∩ B) = N((A ∩ B) ∩ U) = N(U ∩ (A ∩ B)) = N(U ∩ (A ∩ B)c)c = N(U − (A ∩ B)c) = N(U) − N((A ∩ B)c) = N(U) − N(Ac ∪ Bc) = N(U) − N(Ac) + N(Bc) − N(Ac ∩ Bc)

Explore a detailed derivation in set theory for identifying theorems and identities at each step. Learn key concepts like cardinality, intersections, complements, and set difference using the principle of inclusion-exclusion and De Morgan's laws. Ideal for students in undergraduate mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation