JEE Maths›Sets, Relations & Functions›Visual Solution

Visual SolutionHard

Question

The number of onto functions from the set

\{1, 2, 3, 4, 5\}

to the set

\{a, b, c\}

is:

Solution Path

Apply surjection formula with inclusion-exclusion: 3^5 - 3*2^5 + 3*1^5 = 243 - 96 + 3 = 150 onto functions.

01Question Setup

1/4

Find the number of onto functions from

A = \{1, 2, 3, 4, 5\}

B = \{a, b, c\}

. We need every element of

B

to be mapped to.

|A| = 5, \; |B| = 3

02Inclusion-Exclusion Formula

2/4

Use the surjection formula:

\sum_{k=0}^{n} (-1)^k \cdot {}^nC_k \cdot (n-k)^m

where

m = 5

n = 3

{}^3C_0 \cdot 3^5 - {}^3C_1 \cdot 2^5 + {}^3C_2 \cdot 1^5

03Compute Each TermKEY INSIGHT

3/4

243 - 3 \times 32 + 3 \times 1 = 243 - 96 + 3 = 150

. The alternating signs come from inclusion-exclusion.

243 - 96 + 3 = 150

04Final Answer

4/4

Verification: total functions

= 3^5 = 243

, non-onto

= 93

, onto

= 150

\boxed{150}

Concepts from this question2 concepts unlocked

★

Injective, Surjective, and Bijective Functions

STANDARD

A function f: A to B is injective (one-one) if f(a) = f(b) implies a = b. It is surjective (onto) if for every y in B, there exists x in A with f(x) = y. A bijective function is both injective and surjective. For finite sets, if n(A) = n(B), then one-one implies onto and vice versa. The number of one-one functions from A to B (with n(A) <= n(B)) is n(B)!/(n(B)-n(A))!.

\text{One-one: } f(a) = f(b) \Rightarrow a = b \quad | \quad \text{Onto: range}(f) = B

JEE frequently asks students to classify functions as one-one, onto, or bijective. This is the gateway to inverse functions. Understanding these concepts prevents the most common mistake in the chapter: confusing one-one with onto.

Classify functionsCount one-one/onto functionsProve bijectivityInverse function existence

Practice (12 Qs) →

Set Operations and Cardinality

EASY

The fundamental operations on sets are union, intersection, difference, and complement. The inclusion-exclusion principle gives n(A union B) = n(A) + n(B) - n(A cap B). De Morgan's laws relate complements of unions and intersections: (A union B)' = A' cap B' and (A cap B)' = A' union B'. The power set of a set with n elements has 2^n elements.

n(A \cup B) = n(A) + n(B) - n(A \cap B)

Set operations form the foundation of this chapter and appear in almost every JEE question on sets. The inclusion-exclusion principle is frequently tested with 2 or 3 sets. De Morgan's laws are tested both directly and as part of simplification problems.

Cardinality of unionsSet simplificationVenn diagram problemsDe Morgan's law verification

Practice (10 Qs) →

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