Let A = {1, 2, 3} and B = {4, 5, 6}. Find the number of functions from A to B. How many of these functions are one-to-one? How many are onto? Justify your answers.

### Total Number of Functions from \( A \) to \( B \):

A function from set \( A \) to set \( B \) is defined by assigning each element in \( A \) to an element in \( B \). Since \( A \) has 3 elements and \( B \) has 3 elements, for each element in \( A \), we have 3 possible choices from \( B \).

Thus, the total number of functions is calculated as:
\[
|B|^{|A|} = 3^3 = 27.
\]

### Number of One-to-One (Injective) Functions:

For a function to be one-to-one (injective), each element in \( A \) must map to a **distinct** element in \( B \). Therefore, we are selecting 3 distinct elements from \( B \) to assign to the 3 elements in \( A \). This is simply the number of **permutations** of 3 elements from \( B \), which is given by:
\[
P(3, 3) = 3! = 6.
\]

### Number of Onto (Surjective) Functions:

A function is onto (surjective) if every element of \( B \) has at least one preimage in \( A \). To count the number of onto functions from \( A \) to \( B \), we use the inclusion-exclusion principle. The number of functions where all elements of \( B \) are used at least once is given by:
\[
|S_3| = 3! = 6.
\]
This happens because \( A \) and \( B \ can ans

Explore how to count the number of functions from set A = {1, 2, 3} to set B = {4, 5, 6}. Learn how many of these functions are injective (one-to-one) and how many are surjective (onto) using combinatorics, permutations, and the inclusion-exclusion principle. Suitable for high school students in Grades 10-12.

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