Recitation 5: Functions and Sets

Part 1: Sets

True or false:

For the set

$A = \{ \{1, 2, 3\}, \{1, 2\}, \emptyset, \{1\} \},$

Is $1 \in A$ ?
Is $\{ 1, 2 \} \in A$ ?
Is $\{1, 2\} \subseteq A$ ?
Is $\{\emptyset\} \in A$ ?
What is the powerset of $A$ ?
What is the cardinality of the set $\{x \in \mathbb{N} : x \leq 20 \}$ ?
If the cardinality of a set $A$ is five, what is the cardinality of its power set?
What is $A \cap B$ for the sets $A = \{ x \in \mathbb{Z} : 3 \text{ divides } x\}$ and $B = \{ x \in \mathbb{Z} : 4 \text{ divides } x\}$ ?

What is the domain of a function?
What is the target of a function?
What is the range of a function?
Function Validity
- What does it mean for a function to be valid?
- Are these functions valid if their domain is the real numbers? If not, is there a domain that they are valid on?
Injective functions
- What does it mean for a function to be injective?
- Are the above eight functions injective?
- Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 5x + 1$ . Is $f(x)$ injective?
- Let $g: \mathbb{R} \to \mathbb{R}$ be defined by $g(x) = |x|$ . Is $g(x)$ injective?
- what is the horizontal line test and how can we use it on these problems?
Surjective functions
- What does it mean for a function to be surjective?
- Are the above eight functions surjective?
- $h(x) = 5x + 1$
- $j(x) = \sin(x)$
- $k(x) = 2^x$
Bijective functions
- What does it mean for a function to be bijective?
- Which of the previous functions are bijective?