02. Sets & Properties (N, W, Z, Q, Irrational)

Number System me sets (N, W, Z, Q, Irrational, R) aur unki properties: subset/superset, closure, commutative, associative, distributive, identity, inverse; SSC CGL level examples & practice.

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1) Sets kya hote hain? (Hinglish)

Set ka simple meaning: “ek collection” jisme similar type ke elements (numbers) hote hain. Number System me hum numbers ko different groups (sets) me divide karte hain.

2) Important Sets (Definitions + Examples)

Natural Numbers (N)

• Counting numbers: 1, 2, 3, 4, ...
• Note: SSC me generally 0 include nahi hota.

Whole Numbers (W)

• 0 + Natural numbers: 0, 1, 2, 3, ...

Integers (Z)

• Negative + 0 + Positive: ..., -3, -2, -1, 0, 1, 2, 3, ...

Rational Numbers (Q)

• Jo number p/q form me likha ja sake (p, q integers; q ≠ 0).
• All terminating decimals rational hote: 0.125, 2.5, 3.75
• All recurring decimals rational hote: 0.333..., 1.2727...
• Examples: 7/3, -11/2, 0, 5 (5 = 5/1)

Irrational Numbers (Q′)

• Jo p/q form me nahi likhe ja sakte.
• Decimal expansion: non-terminating + non-recurring
• Examples: √2, √3, π

Real Numbers (R)

• All rational + all irrational numbers.
• R = Q ∪ Q′

3) Relationship (subset/superset)

SSC CGL me ye line bahut use hoti hai: N ⊂ W ⊂ Z ⊂ Q ⊂ R

Aur irrational numbers bhi R ke andar aate hain: Q′ ⊂ R.

4) Properties (Very Important for SSC)

Properties ka matlab: operations (+, −, ×, ÷) karne par set ke andar kya behaviour hota hai. Most asked: Closure, Commutative, Associative, Distributive, Identity, Inverse.

A) Closure Property

Meaning: Operation ke baad result usi set me rahe to set “closed” hai.

• N is closed under + and × (generally), but not under − or ÷.
• W closed under + and ×, not under − or ÷.
• Z closed under +, −, ×, not under ÷.
• Q closed under +, −, ×, ÷ (division me denominator 0 nahi hona chahiye).
• Q′ (irrational) is not always closed: e.g. √2 + (−√2) = 0 (rational).

B) Commutative Property

Meaning: a ∘ b = b ∘ a

• Addition & Multiplication: commutative (most sets me)
• Subtraction & Division: not commutative

Example: 7 − 3 ≠ 3 − 7

C) Associative Property

Meaning: (a ∘ b) ∘ c = a ∘ (b ∘ c)

• Addition & Multiplication: associative
• Subtraction & Division: not associative

D) Distributive Property

Meaning: a × (b + c) = ab + ac

Multiplication over addition/subtraction always distributive hota hai.

E) Identity Element

• Additive identity: 0 (a + 0 = a)
• Multiplicative identity: 1 (a × 1 = a)

F) Inverse

• Additive inverse: −a (a + (−a) = 0)
• Multiplicative inverse: 1/a (a × 1/a = 1), where a ≠ 0

Note: Natural/Whole numbers me inverse generally set ke andar nahi rehta (e.g. 2 ka inverse 1/2, N/W me nahi).

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5) Quick Practice (Concept Clear)

1. 2/5 kis set me aata hai? (N/W/Z/Q/Q′/R)
2. 0.666... rational hai ya irrational?
3. √81 aur √8 me kaun rational/irrational?
4. Z (integers) division me closed kyu nahi?
5. Multiplicative identity kya hoti hai?
6. 2 ka additive inverse aur multiplicative inverse likho.

Show Answers

1. Q (and R)
2. Rational (recurring decimal)
3. √81 = 9 rational; √8 irrational
4. Because 1 ÷ 2 = 0.5 (integer nahi)
5. 1 (a × 1 = a)
6. Additive inverse = −2; multiplicative inverse = 1/2