CBSE Class 12 Mathematics Relations and Functions Q #2124
COMPETENCY BASED
APPLY
2 Marks Generic-Curriculum VSA
If $A = \{1, 3, 5, 7\}$ and $B = \{1, 2, 3, 4, 5, 6, 7, 8\}$, then find the number of one-to-one functions from $A$ into $B$.
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Detailed Solution

Step 1: Identify the cardinalities of the sets

The set $A$ contains elements $\{1, 3, 5, 7\}$, so the number of elements in $A$ is $n(A) = 4$. The set $B$ contains elements $\{1, 2, 3, 4, 5, 6, 7, 8\}$, so the number of elements in $B$ is $n(B) = 8$.

Step 2: Apply the formula for one-to-one functions

A function $f: A \to B$ is one-to-one (injective) if each element in $A$ maps to a distinct element in $B$. The number of such functions is given by the permutation formula $P(n, r) = \frac{n!}{(n-r)!}$, where $n = n(B)$ and $r = n(A)$.

Step 3: Calculate the value

Substitute the values into the formula: $$P(8, 4) = \frac{8!}{(8-4)!} = \frac{8!}{4!}$$ $$P(8, 4) = 8 \times 7 \times 6 \times 5 = 1680$$

Final Answer: 1680

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must select and utilize the permutation formula to solve a counting problem related to set mappings.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the application of a specific mathematical algorithm (permutation) to determine the count of functions.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the conceptual understanding of relations and functions combined with basic combinatorial principles, which is a core competency in the Class 12 Mathematics curriculum.

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