Chapter 0 – Quiz 2 – E600 Mathematics

1. For a function $f:X\mapsto\mathbb R$ where $X\subseteq\mathbb R$ , consider the statement $S:= (\forall x\in X: f(x) + 1 \geq f(0))$ . Below, you are given a number of statements and an asserted relationship to $S$ . Select all answers where the assertion is correct. (Multiple choice; Hint: equivalent conditions are also both necessary and sufficient!)

$\forall x\in X: f(x) = 0$ – sufficient

$\nexists x\in X: f(x) < f(0) - 1$ – equivalent

$f(3) \geq f(0) - 1$ – necessary

$\forall x\in X: f(x) = 0$ – equivalent

$\forall x\in X: f(x) \geq f(0)$ – necessary

$\nexists x\in X: f(x) < f(0) - 1$ – sufficient

2. Let $f$ be a function such that $\lim_{x\to 0} f(x) = 0 = f(0)$ . What is false? (Single choice)

The left and right limit of $f$ at $x=0$ may be different from each other.

If $\{x_n\}_{n\in\mathbb N}$ is a sequence with $\lim_{n\to\infty} x_n = 0$ , then $\lim_{n\to\infty} f(x_n) = 0$ .

$f$ is continuous at $x=0$ .

$f$ may be such that $f(x) < 0$ for some elements $x$ in the domain of $f$ .

3. Consider the premise P1: “Every time I watch my sports team play, they lose.” Which pairs of additional premise P2 and conclusion C are such that P1 and P2 imply C? (Multiple choice; Hint/clarification: you may assume that winning and losing are mutually exclusive.)

P2: My sports team has won their latest match; C: I did not watch my sports team’s latest match.

P2: My sports team has lost their latest match; C: I watched my sports team’s latest match.

P2: I watched the last three matches of my sports team; C: My sports team did not win its last two matches.

P2: My sports team has lost their last match; C: I watched all my sports team’s matches this season.

4. Which of the following are true? (Multiple choice)

$\mathbb R\backslash \mathbb N = \varnothing$

$\mathbb N\backslash \mathbb R = \varnothing$

$\{1,2,3\}\cup\{2,3,4\} = \{1,2,2,3,3,4\}$

$[1,4)\cap \mathbb N = \{1,2,3\}$

$\{x\in\mathbb R: x > 2\}^c = (-\infty, 2]$ (complement w.r.t. superset $X=\mathbb R$ )

$\{1,3\}\cap\{2,4\} \neq \varnothing$

5. What is the mathematical nature of the graph of a function $f$ , i.e. what type of object is it?

A set.

A picture.

A function.

A collection of numbers.

6. Consider the conclusion “Snoopy is a good boy”. Which two premises P1 and P2 imply this conclusion? (Single Choice)

P1: All good boys are dogs; P2: Snoopy is a dog.

P1: Some dogs are good boys; P2: Some dogs are named Snoopy.

P1: All dogs are good boys; P2: Snoopy is a dog.

P1: All dogs are good boys; P2: Some dogs are named Snoopy.

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