Thakshashila: Created page with "= De Morgan's Laws - Definition, Explanation, and Examples = '''De Morgan''''s laws are fundamental rules in set theory that describe the relationship between union, intersection, and complements of sets. They help simplify complex set expressions, especially involving complements. == Statements of De Morgan's Laws == Let $A$ and $B$ be two sets and $U$ be the universal set. 1. The complement of the union of two sets is equal to t..."

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Created page with "= De Morgan's Laws - Definition, Explanation, and Examples = '''De Morgan''''s laws are fundamental rules in set theory that describe the relationship between union, intersection, and complements of sets. They help simplify complex set expressions, especially involving complements. == Statements of De Morgan's Laws == Let <math>A</math> and <math>B</math> be two sets and <math>U</math> be the universal set. 1. The complement of the union of two sets is equal to t..."

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= [[De Morgan]]'s Laws - Definition, Explanation, and Examples =

'''De Morgan''''s laws are fundamental rules in set theory that describe the relationship between union, intersection, and complements of sets. They help simplify complex set expressions, especially involving complements.

== Statements of De Morgan's Laws ==

Let <math>A</math> and <math>B</math> be two sets and <math>U</math> be the universal set.

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

<math>
(A \cup B)' = A' \cap B'
</math>

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

<math>
(A \cap B)' = A' \cup B'
</math>

Here, <math>A'</math> denotes the complement of <math>A</math> with respect to <math>U</math>.

== Explanation ==

- The first law means that everything not in either <math>A</math> or <math>B</math> is exactly the elements not in <math>A</math> and not in <math>B</math>.
- The second law means that everything not in both <math>A</math> and <math>B</math> is everything not in <math>A</math> or not in <math>B</math>.

== Examples ==

=== Example 1 ===
Let

<math>U = \{1,2,3,4,5,6\}</math>
<math>A = \{1,2,3\}</math>
<math>B = \{3,4,5\}</math>

Calculate <math>(A \cup B)'</math> and <math>A' \cap B'</math>:

- <math>A \cup B = \{1,2,3,4,5\}</math>
- <math>(A \cup B)' = U - (A \cup B) = \{6\}</math>

Find complements:

- <math>A' = U - A = \{4,5,6\}</math>
- <math>B' = U - B = \{1,2,6\}</math>

Calculate intersection of complements:

- <math>A' \cap B' = \{4,5,6\} \cap \{1,2,6\} = \{6\}</math>

Thus,

<math>(A \cup B)' = A' \cap B' = \{6\}</math>

=== Example 2 ===
Using the same sets,

Calculate <math>(A \cap B)'</math> and <math>A' \cup B'</math>:

- <math>A \cap B = \{3\}</math>
- <math>(A \cap B)' = U - \{3\} = \{1,2,4,5,6\}</math>

Calculate union of complements:

- <math>A' \cup B' = \{4,5,6\} \cup \{1,2,6\} = \{1,2,4,5,6\}</math>

Therefore,

<math>(A \cap B)' = A' \cup B' = \{1,2,4,5,6\}</math>

== Summary ==

* [[De Morgan]]'s laws provide a way to distribute complements over unions and intersections.
* These laws are very useful in simplifying set expressions and solving problems in mathematics, logic, and computer science.

[[Category:Set Theory]]
[[Category:Set Operations]]
[[Category:Mathematics]]

De Morgan’s Laws - Revision history