Set Operations and Important Laws
Data Science and A.I. Lecture Series
Set Operations
Union
Definition:
\[ A \cup B = \{x : x \in A \text{ or } x \in B\} \]
Examples:
- \( A = \{1, 2, 3\}, B = \{3, 4, 5\} \implies A \cup B = \{1, 2, 3, 4, 5\} \)
- \( A = \{a, b\}, B = \{b, c, d\} \implies A \cup B = \{a, b, c, d\} \)
Intersection
Definition:
\[ A \cap B = \{x : x \in A \text{ and } x \in B\} \]
Examples:
- \( A = \{1, 2, 3\}, B = \{3, 4, 5\} \implies A \cap B = \{3\} \)
- \( A = \{a, b, c\}, B = \{b, c, d\} \implies A \cap B = \{b, c\} \)
Complement
Definition:
\[ A^c = \{x \in U : x \notin A\} \]
Examples:
- \( U = \{1, 2, 3, 4, 5\}, A = \{1, 2\} \implies A^c = \{3, 4, 5\} \)
- \( U = \{a, b, c, d\}, A = \{a, b\} \implies A^c = \{c, d\} \)
Important Laws of Sets
Idempotent and Identity Laws
Idempotent Laws:
\[ A \cup A = A, \quad A \cap A = A \]
Identity Laws:
\[ A \cup \emptyset = A, \quad A \cap U = A \]
Example:
\( A = \{1, 2\}, U = \{1, 2, 3\}, \emptyset = \{\} \)
\( A \cup \emptyset = \{1, 2\}, \quad A \cap U = \{1, 2\} \)
Distributive and De Morgan’s Laws
Distributive Laws:
\[ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \]
\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]
De Morgan’s Laws:
\[ (A \cup B)^c = A^c \cap B^c \]
\[ (A \cap B)^c = A^c \cup B^c \]
Applications of Sets
Practical problems using sets:
- \( n(A \cup B) = n(A) + n(B) – n(A \cap B) \)
- \( n(A \cup B) = n(A – B) + n(A \cap B) + n(B – A) \)
Example: Language Proficiency
In a group of 500 persons, 400 can speak Hindi and 150 can speak English. Using formulas, the results are:
- Both Hindi and English: \( 50 \)
- Only Hindi: \( 350 \)
- Only English: \( 100 \)
Video
Thank You!
Follow PostNetwork Academy for more:
- Website: www.postnetwork.co
- YouTube: PostNetwork Academy