练习

这里有一组对象：🍎 🍌 🍐。下面哪一个是这一组的成员？

这一组有几个成员？🍊 🥕 🍋 🍎 🍐

健二的盒子里有 🍎 🍌 🍐。露西娅的盒子里有 🍐 🍎 🍌。它们是同一组吗？

装蓝色水果的盒子是空的。空组有几个成员？

Gideon 的游戏：这一组是 🍌 🍋 🥒。下面的说法是真还是假？“这一组的每个成员都是黄色的。”

给大一点的孩子：从苹果（apple）、香蕉（banana）、草莓（strawberry）和柠檬（lemon）中，写出由红色东西组成的一组。请使用代码中显示的英文单词，用逗号分隔，例如 {kiwi, plum}。

Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. Write $A \cup B$.

With the same $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$, write $A \cap B$.

In the universe $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, write the complement $A^{c}$ of $A = \{1, 2, 3, 4\}$.

Which statements are true? (Choose all that apply.)

What is the cardinality $|\{a, b, a, c, b\}|$?

A café offers 3 drinks and 4 snacks. A combo is an ordered pair (drink, snack). If $D$ is the set of drinks and $S$ the set of snacks, how many elements does $D \times S$ have?

In a class, 14 students play football, 9 play chess, and 5 play both. How many play at least one of the two games?

Write a Python function is_subset(a, b) that returns True exactly when every element of the set a is an element of the set b. Don't use Python's built-in <= or issubset — loop over the elements instead.

Compute $-8 + 5$.

Compute $-4 - (-9)$.

Compute $(-3) \cdot (-4) \cdot (-1)$.

Compute $(-6) + (-7)$.

In the proof that $(-a)(-b) = ab$, we wrote $(-a)(-b) + (-a)b = (-a)\big((-b) + b\big)$. Which property justifies this step?

For which integers $a$ is there an integer $b$ with $a \cdot b = 1$? (Choose all that apply.)

At dawn in Ulaanbaatar the temperature is $-23\,^\circ\mathrm{C}$. By noon it has risen by $15$ degrees. What is the noon temperature, in $^\circ\mathrm{C}$?

Write find_mult_partner(a, limit) that searches the integers $b$ with $-\text{limit} \le b \le \text{limit}$ and returns the first $b$ with a * b == 1, or None if there is none. Then think: why does searching further never help when $|a| \ge 2$?

Let $A = \{1, 2, 3\}$ and $B = \{a, b\}$. Which of these rules is a function from $A$ to $B$?

How many functions are there from a set with $3$ elements to a set with $2$ elements?

How many injective functions are there from a 2-element set to a 4-element set?

Which of these functions $\Z \to \Z$ are surjective? (Choose all that apply.)

A school has 400 students. Why must at least two of them share a birthday (day and month)?

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.

Let $f(x) = x + 3$ and $g(x) = x^2$. Compute $(g \circ f)(2)$.

If $f : A \to B$ and $g : B \to C$, what are the domain and codomain of $g \circ f$?

Let $f(x) = 3x$ and $g(x) = x - 2$. Compute $(g \circ f)(4) - (f \circ g)(4)$.

Suppose $g \circ f$ is injective. Which of these must be true? (Choose all that apply.)

In Lean, comp_assoc is proved by rfl. What does that tell us?

Finite functions can be stored as dictionaries. Write compose(g, f) returning a dictionary for $g \circ f$ (apply f first!). You may assume every value of f is a key of g.