1. Set Theory Foundations

Definition 1.1: Set

A set is a well-defined collection of distinct objects. A set is:

Well-defined: For any object, we can determine if it belongs to the set
Distinct: Each element appears at most once
Unordered: Order does not matter

Definition 1.2: Set Operations

For sets A and B:

$A \cup B = \{x : x \in A \text{ or } x \in B\}$ (union)
$A \cap B = \{x : x \in A \text{ and } x \in B\}$ (intersection)
$A \setminus B = \{x : x \in A \text{ and } x \notin B\}$ (difference)
$A^c = \{x \in U : x \notin A\}$ (complement)

Theorem 1.1: De Morgan's Laws

For sets A and B in universal set U:

(A \cup B)^c = A^c \cap B^c

(A \cap B)^c = A^c \cup B^c

Proof:

We prove the first law by double inclusion:

Part 1: Let $x \in (A \cup B)^c$ . Then $x \notin A \cup B$ , so $x \notin A$ and $x \notin B$ . Therefore $x \in A^c \cap B^c$ .

Part 2: Let $x \in A^c \cap B^c$ . Then $x \in A^c$ and $x \in B^c$ , so $x \notin A$ and $x \notin B$ . Therefore $x \notin A \cup B$ , so $x \in (A \cup B)^c$ .

∎

Example 1.1: Set Operations

Let $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$ .

$A \cup B = \{1, 2, 3, 4\}$
$A \cap B = \{2, 3\}$
$A \setminus B = \{1\}$

2. Properties of Real Numbers

Definition 2.1: Bounded Set

A set $S \subseteq \mathbb{R}$ is:

Bounded above if there exists $M \in \mathbb{R}$ such that $x \leq M$ for all $x \in S$
Bounded below if there exists $m \in \mathbb{R}$ such that $m \leq x$ for all $x \in S$
Bounded if it is bounded above and below

Definition 2.2: Supremum and Infimum

For a set $S \subseteq \mathbb{R}$ :

Supremum (sup S): The least upper bound of S
Infimum (inf S): The greatest lower bound of S

If sup S ∈ S, then sup S = max S. Otherwise, max S does not exist.

Theorem 2.1: Completeness Property

Every non-empty set of real numbers that is bounded above has a supremum in $\mathbb{R}$ .

Example 2.1: Finding Supremum

For $S = \{x \in \mathbb{Q} : x^2 < 2\}$ :

sup S = $\sqrt{2}$ (in $\mathbb{R}$ )
max S does not exist (since $\sqrt{2} \notin \mathbb{Q}$ )

3. Functions Fundamentals

Definition 3.1: Function

A function $f: A \to B$ is a rule that assigns to each element $x \in A$ exactly one element $f(x) \in B$ .

Domain: The set A
Codomain: The set B
Range: $\{f(x) : x \in A\}$

Definition 3.2: Bounded Function

A function $f: D \to \mathbb{R}$ is bounded if there exists $M > 0$ such that $|f(x)| \leq M$ for all $x \in D$ .

Definition 3.3: Monotonic Functions

A function $f: I \to \mathbb{R}$ is:

Increasing if $f(x_1) \leq f(x_2)$ whenever $x_1 < x_2$
Strictly increasing if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$
Decreasing if $f(x_1) \geq f(x_2)$ whenever $x_1 < x_2$

Example 3.1: Function Composition

Let $f(x) = x^2$ and $g(x) = x + 1$ .

$(f \circ g)(x) = f(g(x)) = f(x+1) = (x+1)^2$
$(g \circ f)(x) = g(f(x)) = g(x^2) = x^2 + 1$

4. Real Number Construction

Definition 4.1: Dedekind Cut

A Dedekind cut is a partition of $\mathbb{Q}$ into two non-empty sets $A$ and $B$ such that:

Every rational is in exactly one of A or B
If $a \in A$ and $b \in B$ , then $a < b$
A has no greatest element

Theorem 4.1: Completeness Axiom

Every non-empty set of real numbers that is bounded above has a supremum in $\mathbb{R}$ .

This property distinguishes $\mathbb{R}$ from $\mathbb{Q}$ .

Example 4.1: Irrational Number via Dedekind Cut

The number $\sqrt{2}$ is defined as the Dedekind cut:

A = \{r \in \mathbb{Q} : r < 0 \text{ or } r^2 < 2\}, \quad B = \{r \in \mathbb{Q} : r > 0 \text{ and } r^2 > 2\}

This cut has no rational supremum, so it represents an irrational number.

5. Inequalities and Properties

Theorem 5.1: Triangle Inequality

For any real numbers $x, y$ :

|x + y| \leq |x| + |y|

Proof:

Since $-|x| \leq x \leq |x|$ and $-|y| \leq y \leq |y|$ , adding gives:

-(|x| + |y|) \leq x + y \leq |x| + |y|

Therefore $|x + y| \leq |x| + |y|$ .

∎

Theorem 5.2: Archimedean Property

For any real number $x > 0$ , there exists a natural number $n$ such that $n > x$ .

Example 5.1: Density of Rationals

Theorem: Between any two distinct real numbers, there exists a rational number.

Proof sketch: Use the Archimedean property to find a rational with denominator large enough to fit between the two reals.

6. Function Composition and Inverse

Definition 6.1: Function Composition

If $f: B \to C$ and $g: A \to B$ , then the composition $f \circ g: A \to C$ is defined by:

(f \circ g)(x) = f(g(x))

Definition 6.2: Injective Function

A function $f: A \to B$ is injective (one-to-one) if:

f(x_1) = f(x_2) \Rightarrow x_1 = x_2

Definition 6.3: Surjective Function

A function $f: A \to B$ is surjective (onto) if for every $y \in B$ , there exists $x \in A$ such that $f(x) = y$ .

Definition 6.4: Inverse Function

If $f: A \to B$ is bijective (both injective and surjective), then the inverse function $f^{-1}: B \to A$ satisfies:

f^{-1}(f(x)) = x \text{ and } f(f^{-1}(y)) = y

Theorem 6.1: Inverse Function Theorem

If $f$ is strictly monotonic on an interval $I$ , then $f$ is injective and has an inverse $f^{-1}$ on $f(I)$ .

Example 6.1: Composition Properties

Let $f(x) = x^2$ and $g(x) = x + 1$ .

$(f \circ g)(x) = (x+1)^2 = x^2 + 2x + 1$
$(g \circ f)(x) = x^2 + 1$
Note: $f \circ g \neq g \circ f$ in general

7. Number Sets and Cardinality

Definition 7.1: Countable Set

A set $A$ is countable if there exists a bijection $f: \mathbb{N} \to A$ . A set is uncountable if it is not countable.

Theorem 7.1: Countability Results

$\mathbb{N}, \mathbb{Z}, \mathbb{Q}$ are countable
$\mathbb{R}$ is uncountable (Cantor's diagonal argument)
The power set $\mathcal{P}(A)$ has cardinality $2^{|A|}$

Example 7.1: Cardinality Examples

$|\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}| = \aleph_0$ (countably infinite)
$|\mathbb{R}| = 2^{\aleph_0} = c$ (continuum)
$|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0} = c$

Frequently Asked Questions

What's the difference between ⊆ and ⊂?

A ⊆ B (subset) means every element of A is in B, including the case A = B. A ⊂ B (proper subset) means A ⊆ B AND A ≠ B.

Why is the empty set a subset of every set?

By definition, A ⊆ B means 'for all x, if x ∈ A then x ∈ B'. For the empty set ∅, there are no elements, so the statement is vacuously true for any B.

What is supremum and how is it different from maximum?

The supremum (sup) of a set S is the least upper bound. It may or may not be in S. The maximum is the largest element in S, so it must be in S. For example, sup(0,1) = 1 but max(0,1) doesn't exist.

How do I prove two sets are equal?

Prove double inclusion: show A ⊆ B and B ⊆ A. This is called the 'element-chasing' method.

What makes a function bounded?

A function f is bounded on a set D if there exists M > 0 such that |f(x)| ≤ M for all x ∈ D. This means the function values stay within a finite range.

Practice Quiz

Real Numbers and Functions Practice

10

Questions

0

Correct

0%

Accuracy

1

If

A = \{1, 2, 3\}

and

B = \{2, 3, 4\}

, what is

A \cap B

?

Easy

Not attempted

2

According to De Morgan's Law,

(A \cup B)^c

equals:

Medium

Not attempted

3

What is the supremum of the set

S = \{x \in \mathbb{Q} : x^2 < 2\}

?

Hard

Not attempted

4

If

f(x) = x^2

and

g(x) = x + 1

, what is

(f \circ g)(x)

?

Easy

Not attempted

5

Which of the following sets is countably infinite?

Medium

Not attempted

6

A function

f

is strictly increasing if:

Easy

Not attempted

7

What is the cardinality of the power set of

\{a, b, c\}

?

Easy

Not attempted

8

If

A \subseteq B

and

B \subseteq A

, what can we conclude?

Easy

Not attempted

9

The completeness property of

\mathbb{R}

states that:

Medium

Not attempted

10

If

f(x) = |x|

, is

f

bounded on

\mathbb{R}

?

Medium

Not attempted