The Lumenary · 6.º a 8.º grado

Functions

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1.
Let $A = \{1, 2, 3\}$ and $B = \{a, b\}$ . Which of these rules is a function from $A$ to $B$ ?
- A. $1 \mapsto a,\; 2 \mapsto a,\; 3 \mapsto b$
- B. $1 \mapsto a,\; 2 \mapsto b$
- C. $1 \mapsto a,\; 1 \mapsto b,\; 2 \mapsto a,\; 3 \mapsto b$
2.
How many functions are there from a set with $3$ elements to a set with $2$ elements?
3.
How many injective functions are there from a 2-element set to a 4-element set?
4.
Which of these functions $\Z \to \Z$ are surjective? (Choose all that apply.)
- A. $n \mapsto n + 1$
- B. $n \mapsto 2n$
- C. $n \mapsto -n$
- D. $n \mapsto n^2$
5.
A school has 400 students. Why must at least two of them share a birthday (day and month)?
- A.There are at most 366 possible birthdays, so the function from students to birthdays cannot be injective.
- B.Because birthdays are random.
- C.It isn't certain; it's just very likely.
6.
A finite function can be stored as a Python dictionary, e.g. {1: 'a', 2: 'b', 3: 'a'}. Write is_injective(f) returning True exactly when no two keys have the same value.
Computacional — Práctica

Solucionario — Functions

1.
Only the first rule gives every element of $A$ exactly one output. The second forgets $3$ ; the third sends $1$ to two places.
2.
Each of the 3 inputs has 2 possible outputs, independently, so there are $2^3 = 8$ functions. In general there are $|B|^{|A|}$ functions $A \to B$ .
3.
The first input has $4$ choices and the second $3$ , so there are $4 \cdot 3 = 12$ injective functions.
4.
$n \mapsto n+1$ and $n \mapsto -n$ are surjective (solve $n = m - 1$ and $n = -m$ ). $n \mapsto 2n$ misses every odd number, and $n \mapsto n^2$ misses every negative number.
5.
The function $\text{students} \to \text{dates}$ goes from a set of size 400 to a set of size at most 366. An injective function cannot map a larger finite set into a smaller one, so two students must share a date. This is the pigeonhole principle — no probability required.
6.
```
def is_injective(f):
    return len(set(f.values())) == len(f)
```
Putting the outputs in a set removes repeats. If no output is repeated, the set has one element per input — exactly the definition of injectivity for a finite function.