How Many Different Function are Possible

Duration: 3 min

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AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This video lecture explains the formula for calculating the number of possible functions from one set to another. The instructor presents a problem: given sets A and B with cardinalities |A| = m and |B| = n, determine the number of functions from A to B. He underlines the cardinalities and writes the answer $n^m$. He uses a concrete example. He defines set A with elements 'a' and 'b' and set B with '1', '2', '3'. He demonstrates the counting principle. For each element in the domain, there are 'n' choices in the codomain. Since there are 'm' elements in the domain, the total is $n^m$.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor introduces the problem statement on the slide: "If |A| = m and |B| = n, then number of functions possible from A to B = ?". He underlines the cardinalities and writes the answer $n^m$. He draws two sets, A and B. He populates set A with 'a' and 'b' and set B with '1', '2', '3'. He writes |A| = m = 2 and |B| = n = 3. He writes "x3" next to 'a'. He writes another "x3" next to 'b'. He explains the choices for mapping. This section establishes the specific example used to derive the general formula.

  2. 2:00 – 2:34 02:00-02:34

    The instructor completes the calculation. He writes "3 x 3 = 9" to show the total number of functions for the specific case where m=2 and n=3. He writes $3^2$ and circles it, connecting the numerical result back to the general formula $n^m$. He circles the $n^m$ on the original slide text to reinforce the final answer. He emphasizes the relationship between the base and exponent. This concludes the derivation.

The lecture bridges abstract set theory notation and concrete counting principles. By starting with the general formula $n^m$ and grounding it in a specific example with sets of size 2 and 3, the instructor makes the concept accessible. The visual aid of drawing arrows and writing "x3" next to each element in set A clearly demonstrates the multiplication principle of counting. The final step of circling $n^m$ ties the specific calculation (3x3=9) back to the general rule, ensuring students understand that the exponent 'm' represents the number of elements in the domain (set A) and the base 'n' represents the number of elements in the codomain (set B). This ensures a clear understanding of the formula's application.

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