Number of Injective Functions are Possible

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This educational video segment explains the mathematical principles for determining the number of possible functions between two sets, specifically focusing on one-to-one mappings. The instructor begins by presenting the standard permutation formula, $^nP_m = P(n, m)$, as the general method for counting these functions. He highlights a specific scenario where the cardinality of both sets is equal ($|A| = |B| = n$), resulting in $n!$ possible functions. To illustrate the general case, he introduces variables $m$ and $n$ for the sizes of sets A and B respectively. He then constructs a visual example on the whiteboard, defining set A with five distinct elements (1, 2, 3, 4, 5) and set B with six distinct elements (a, b, c, d, e, f). He demonstrates the concept of a function by drawing arrows from elements in set A to elements in set B, showing how each element in the domain must map to exactly one element in the codomain.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The instructor starts by writing the formula for the number of possible functions as $^nP_m = P(n, m)$. He immediately follows this with a special case: if the size of set A equals the size of set B ($|A| = |B| = n$), then the number of functions possible is $n!$. He then begins a practical demonstration by writing $|A| = m$ and $|B| = n$. He draws two vertical ovals representing the sets. Inside the first oval, labeled A, he lists the numbers 1 through 5. Inside the second oval, labeled B, he lists the letters a through f. He draws arrows originating from the elements of set A pointing towards set B to visualize the mapping process. Below the diagram, he writes out the multiplication sequence $6 imes 5 imes 4 imes 3 imes 2$, representing the number of choices available for mapping each element of A to B.

2. 2:00 – 3:47 02:00-03:47

   Continuing the example, the instructor writes the formal permutation formula $^nP_m = rac{n!}{(n-m)!}$ on the board. He substitutes the specific values from his example, writing $^6P_5 = rac{6!}{(6-5)!}$. He simplifies the denominator to $1!$ and then to $1$, leaving the result as $6!$. He expands $6!$ into its full product form: $6 imes 5 imes 4 imes 3 imes 2 imes 1$. This confirms the manual calculation he performed earlier in the diagram section. He concludes the segment by summarizing the relationship between the set sizes, writing $|A| = m ightarrow |B| = n$ to reinforce the variables used in the permutation formula for one-to-one functions.

The lecture effectively bridges the gap between abstract formulas and concrete counting methods. By starting with the general permutation formula and then applying it to a specific numerical example, the instructor clarifies how to calculate the number of one-to-one functions. The visual aid of drawing sets and arrows helps students understand the underlying logic of the multiplication principle used in the formula. The final expansion of the factorial confirms the consistency between the algebraic formula and the step-by-step counting method.