De Morgan's Laws: A Practical Guide to Set Theory

What are De Morgan’s Laws?

De Morgan’s Laws are two fundamental principles in set theory that describe how the intersection and union of sets relate to their complements. The laws are as follows:

1. The complement of the union of two sets is equal to the intersection of their complements.

   Mathematically:
   \[ (A \cup B)’ = A’ \cap B’ \]

2. The complement of the intersection of two sets is equal to the union of their complements.

   Mathematically:
   \[ (A \cap B)’ = A’ \cup B’ \]

Let’s break these down with practical examples.

Example 1: The Union of Two Sets

Given:

• Set A: {1, 2, 3}
• Set B: {3, 4, 5}

Step 1: Find the Union of A and B

• Union (A ∪ B): {1, 2, 3, 4, 5}

Step 2: Find the Complement of the Union

• Universal Set (U): {1, 2, 3, 4, 5, 6, 7}
• Complement (A ∪ B)’: {6, 7}

Step 3: Find the Complements of A and B

• A’: {4, 5, 6, 7}
• B’: {1, 2, 6, 7}

Step 4: Find the Intersection of A’ and B’

• Intersection (A’ ∩ B’): {6, 7}

Conclusion:

• You can see that (A ∪ B)’ = A’ ∩ B’, which confirms De Morgan’s first law.

Example 2: The Intersection of Two Sets

Given:

• Set C: {1, 2, 3}
• Set D: {2, 3, 4}

Step 1: Find the Intersection of C and D

• Intersection (C ∩ D): {2, 3}

Step 2: Find the Complement of the Intersection

• Complement (C ∩ D)’: {1, 4, 5, 6, 7}

Step 3: Find the Complements of C and D

• C’: {4, 5, 6, 7}
• D’: {1, 5, 6, 7}

Step 4: Find the Union of C’ and D’

• Union (C’ ∪ D’): {1, 4, 5, 6, 7}

Conclusion:

• You can see that (C ∩ D)’ = C’ ∪ D’, confirming De Morgan’s second law.

Summary

De Morgan’s Laws provide a powerful framework for understanding the relationships between sets and their complements. By practicing with these laws using concrete examples, you can strengthen your grasp of set theory and its applications in logic and mathematics.