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:

• $A=\{1,2,3\},B=\{3,4,5\}⟹A\cup B=\{1,2,3,4,5\}$
• $A=\{a,b\},B=\{b,c,d\}⟹A\cup B=\{a,b,c,d\}$

Intersection

Definition:

$$A\cap B=\{x:x\in A\text{ and }x\in B\}$$

Examples:

• $A=\{1,2,3\},B=\{3,4,5\}⟹A\cap B=\{3\}$
• $A=\{a,b,c\},B=\{b,c,d\}⟹A\cap B=\{b,c\}$

Complement

Definition:

$$A^c=\{x\in U:x\notin A\}$$

Examples:

• $U=\{1,2,3,4,5\},A=\{1,2\}⟹A^c=\{3,4,5\}$
• $U=\{a,b,c,d\},A=\{a,b\}⟹A^c=\{c,d\}$

Important Laws of Sets

Idempotent and Identity Laws

Idempotent Laws:

$$A\cup A=A,A\cap A=A$$

Identity Laws:

$$A\cup \mathrm{\varnothing }=A,A\cap U=A$$

Example:

$A=\{1,2\},U=\{1,2,3\},\mathrm{\varnothing }=\{\}$

$A\cup \mathrm{\varnothing }=\{1,2\},A\cap U=\{1,2\}$

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: $100$

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