Relations & Functions — Exercises

Contents

1Relation as a set of tuples easy

Let $A = \{1, 2, 3\}$ and $B = \{a, b, c\}$. Define the relation $R = \{(1, a), (1, b), (2, c), (3, b)\}$.

1. Verify that $R \subseteq A \times B$.
2. List all elements in $A$ that have at least one image in $R$ (the domain of definition).
3. List all elements in $B$ that appear as an image (the image set).

2Is this a function? easy

For each relation on $A = \{1, 2, 3, 4\}$, decide whether it is a function. Use Python.

1. $R_1 = \{(1, 1), (2, 4), (3, 9), (4, 16)\}$
2. $R_2 = \{(1, 1), (1, 2), (2, 4), (3, 9), (4, 16)\}$
3. $R_3 = \{(1, 1), (2, 4), (3, 9)\}$ — domain still $A = \{1,2,3,4\}$

3Domain & range (finite) easy

Given the dictionary f = {1:'a', 2:'b', 3:'a', 4:'c', 5:'b'}, compute the domain, the image, and the preimage of 'a' and of 'b'.

4Domain of $\dfrac{\sqrt{x-3}}{x-7}$ medium

Find the (real) domain of $f(x) = \dfrac{\sqrt{x-3}}{x-7}$, both by hand and with SymPy.

5Injectivity test medium

Decide whether each of these is injective on its domain:

1. $f_1(x) = 3x - 4$ on $\mathbb{R}$
2. $f_2(x) = x^2 - 4x + 7$ on $\mathbb{R}$
3. $f_3(x) = x^2 - 4x + 7$ on $[2, \infty)$
4. $f_4 = \{1\!\to\!a,\, 2\!\to\!b,\, 3\!\to\!a\}$

6Surjectivity test medium

Is the function $f : \mathbb{R} \to \mathbb{R},\; f(x) = e^x$ surjective? What about $f : \mathbb{R} \to (0, \infty),\; f(x) = e^x$?

7Inverse of a finite bijection medium

Let $f = \{1\!\to\!v,\, 2\!\to\!x,\, 3\!\to\!w,\, 4\!\to\!z\}$ from $\{1,2,3,4\}$ to $\{v,w,x,z\}$. Build $f^{-1}$ and verify $f^{-1}(f(k)) = k$ for every key.

8Symbolic inverse medium

Find the inverse of $f(x) = 2x + 5$ and of $g(x) = \dfrac{x-1}{x+1}$. Verify with SymPy.

9Composition medium

Let $f(x) = x^2 + 1$ and $g(x) = 2x - 3$. Compute $f \circ g$ and $g \circ f$ symbolically with SymPy, then verify numerically that $f(g(2)) = 2$ and $g(f(2)) = 7$.

10Pipe of $n$ functions hard

Write pipe(*fs) such that pipe(f1, f2, …, fn)(x) = fn(…f2(f1(x))) (left-to-right). Then write compose(*fs) for the mathematical right-to-left convention (f1(f2(…fn(x)))).

11$x^2$ vs $x^3$ — invertibility hard

Decide whether each is invertible on its stated domain, and give the inverse when it exists.

1. $f(x) = x^2$ on $\mathbb{R}$
2. $f(x) = x^2$ on $[0, \infty)$
3. $f(x) = x^3$ on $\mathbb{R}$

12Sigmoid example hard

Let $f(x) = \dfrac{-x}{-1+x}$ and $g(x) = \log(x)$. Compute $h = f^{-1} \circ g^{-1}$, then find its domain and range with SymPy. Recognize the result.

Going further

• Implement iterate(f, n) that returns $f^n = f \circ f \circ \cdots \circ f$ ($n$ times).
• For a random injection $f : \{1,\dots,10\} \to \{1,\dots,10\}$, verify $f \circ f^{-1} = \mathrm{id}$.
• With SymPy, prove $\log$ and $\exp$ are inverses on $(0, \infty)$.
• Visualize the sigmoid from exercise 12 with matplotlib.

Open the main lesson for the matching theory.