To solve the problem of finding the total number of "into" functions from set $X$ to set $Y$, let's first define what an "into" function means and analyze the given sets.

Definitions:

• Into Function: A function $f: X \to Y$ is an "into" function if not all elements of $Y$ have a pre-image in $X$, meaning that some elements of $Y$ are not mapped to by any element of $X$.
• In contrast, an "onto" function would ensure that every element of $Y$ has at least one pre-image in $X$.

Given:

• The set $X$ has 3 elements.
• The set $Y$ has 2 elements.

Analysis:

We want to find the total number of functions from $X$ to $Y$ that are "into" functions, i.e., not every element in $Y$ is mapped by elements in $X$.

1. Total Number of Functions:

• Since $X$ has 3 elements and $Y$ has 2 elements, each element in $X$ can map to any of the 2 elements in $Y$.
• The total number of functions from $X$ to $Y$ is $2^3 = 8$.

2. Number of Onto Functions:

• An onto function would map all elements in $X$ such that every element in $Y$ has at least one pre-image.

• The number of onto functions from a set of size 3 to a set of size 2 can be found using the formula for the number of onto functions: $\text{Number of onto functions} = m! \times S(n, m)$ where $S(n, m)$ is the Stirling number of the second kind, representing the number of ways to partition a set of $n$ elements into $m$ non-empty subsets, and $m!$ is the factorial of $m$.

  For $n = 3$ and $m = 2$:

  • $S(3, 2) = 3$ (3 ways to partition 3 elements into 2 non-empty subsets).
  • Therefore, the number of onto functions is: $2! \times 3 = 2 \times 3 = 6$

3. Number of Into Functions:

• Since into functions are those that are not onto, the number of into functions is: $\text{Total functions} - \text{Onto functions} = 8 - 6 = 2$

Conclusion:

The total number of into functions from $X$ to $Y$ is 2.

Would you like any details or have any questions?

Relative Questions:

1. What is the difference between an "into" and "onto" function?
2. How is the Stirling number of the second kind used in counting onto functions?
3. How would the number of into functions change if $X$ had 4 elements and $Y$ had 3 elements?
4. What are the general properties of functions between two finite sets?
5. How do you calculate the number of functions between sets with more elements?

Tip:

When counting specific types of functions like "into" or "onto," it's crucial to first determine the total number of possible functions, and then apply relevant counting principles or formulas to narrow down to the desired category.