CBSE Class 12 Mathematics Relations and Functions Q #2123
COMPETENCY BASED
APPLY
2 Marks Generic-Curriculum VSA
Find the number of surjections from $A = \{1, 2, \dots, n\}, n \ge 2$, onto $B = \{a, b\}$.
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Detailed Solution

Step 1: Understand the definition of a surjection

A function $f: A \to B$ is surjective (onto) if every element in the codomain $B$ has at least one pre-image in the domain $A$. Here, $|A| = n$ and $|B| = 2$.

Step 2: Calculate total functions

The total number of functions from set $A$ to set $B$ is given by $|B|^{|A|}$. Since $|A| = n$ and $|B| = 2$, the total number of functions is $2^n$.

Step 3: Identify non-surjective functions

A function is not surjective if its range is a proper subset of $B$. Since $B = \{a, b\}$, the only non-surjective functions are those where the range is either $\{a\}$ or $\{b\}$. These are the constant functions.

Step 4: Subtract non-surjective functions

There are exactly 2 constant functions: one mapping all elements to $a$, and one mapping all elements to $b$. Thus, the number of surjections is: $$2^n - 2$$

Final Answer: 2^n - 2

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the combinatorial principle of inclusion-exclusion or the complement method to determine the count of surjective mappings.
Knowledge Dimension: PROCEDURAL
Justification: The student follows a specific algorithmic process (Total functions - Constant functions) to arrive at the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This tests the conceptual understanding of Relations and Functions beyond rote memorization of definitions.

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