Sets, Relations, and Functions

Basic operations:

• Union: $A \cup B = \\{x : x \in A \mathrm{ or} x \in B\\}$
• Intersection: $A \cap B = \\{x : x \in A \mathrm{ and} x \in B\\}$
• Difference: $A \setminus B = \\{x : x \in A \mathrm{ and} x \notin B\\}$
• Complement: $A^c = U \setminus A$ (where $U$ is the universal set)

De Morgan”s Laws:

$(A \cup B)^c = A^c \cap B^c, \quad (A \cap B)^c = A^c \cup B^c$

Power set: $\mathcal{P}(A) = \\{B : B \subseteq A\\}$. If $|A| = n$Then $|\mathcal{P}(A)| = 2^n$.

A binary relation $R$ from set $A$ to set $B$ is a subset of $A \times B$.

A relation $R$ on $A$ is:

• Reflexive: $\forall a \in A$, $(a,a) \in R$.
• Symmetric: $(a,b) \in R \implies (b,a) \in R$.
• Antisymmetric: $(a,b) \in R$ and $(b,a) \in R \implies a = b$.
• Transitive: $(a,b) \in R$ and $(b,c) \in R \implies$(a,c) \in R$.

Equivalence relation: reflexive, symmetric, transitive. Partitions the set into equivalence classes.

Partial order: reflexive, antisymmetric, transitive. Written $(A, \preceq)$.

A Hasse diagram is a graphical representation of a finite poset $(A, \preceq)$: an element $a$ Is drawn below $b$ whenever $a \prec b$ (i.e., $a \preceq b$ and $a \neq b$), and an edge is drawn From $a$ to $b$ whenever $b$ covers $a$ (there is no $c$ with $a \prec c \prec b$).

Worked Example. Show that $R$ on $\mathbb{{'}Z{}'}$ defined by $a\,R\,b$ iff $a \equiv b \pmod{5}$ is An equivalence relation. Describe the equivalence classes.

Worked Example. Let $\preceq$ be divisibility on $A = \\{1, 2, 3, 4, 6, 12\\}$: $a \preceq b$ iff $a \mid b$. Verify this is a partial order and identify the cover relations.

A function $f : A \to B$ is a relation where each $a \in A$ appears exactly once as a first element.

• Injective (one-to-one): $f(a_1) = f(a_2) \implies a_1 = a_2$.
• Surjective (onto): for every $b \in B$There exists $a \in A$ with $f(a) = b$.
• Bijective: both injective and surjective.

Theorem 2.1. If $A$ and $B$ are finite sets, $f : A \to B$ is:

• Injective if and only if $|A| \leq |B|$.
• Surjective if and only if $|A| \geq |B|$.
• Bijective if and only if $|A| = |B|$.

Theorem 2.2 (Pigeonhole Principle). If $|A| \gt{} |B|$Then no function $f : A \to B$ is injective. Equivalently, placing $n$ items into $m$ boxes with $n \gt{} m$ forces at least one box to contain at least $\lceil n/m \rceil$ items.

Function composition. Given $f : A \to B$ and $g : B \to C$The composition $g \circ f : A \to C$ is defined by $(g \circ f)(a) = g(f(a))$ for all $a \in A$.

Theorem 2.3. If $f : A \to B$ and $g : B \to C$ are both injective, then $g \circ f$ is injective.

Proof. Suppose $(g \circ f)(a_1) = (g \circ f)(a_2)$. Then $g(f(a_1)) = g(f(a_2))$. Since $g$ is Injective, $f(a_1) = f(a_2)$. Since $f$ is injective, $a_1 = a_2$. Hence $g \circ f$ is injective. $\blacksquare$

Theorem 2.4. If $f : A \to B$ and $g : B \to C$ are both surjective, then $g \circ f$ is surjective.

Proof. Let $c \in C$. Since $g$ is surjective, $\exists\, b \in B$ with $g(b) = c$. Since $f$ is Surjective, $\exists\, a \in A$ with $f(a) = b$. Then $(g \circ f)(a) = g(f(a)) = g(b) = c$. Hence $g \circ f$ is surjective. $\blacksquare$

Corollary 2.5. The composition of two bijections is a bijection.

A function $f : A \to B$ is invertible if there exists $f^{-1} : B \to A$ such that f^{-1} \circ f = \mathrm{id{}_A and f \circ f^{-1} = \mathrm{id{}_B. A function is invertible If and only if it is bijective.

Definition. A set $S$ is countable if it is finite or countably infinite. A set is countably infinite if there exists a bijection $\mathbb{{'}N{}'} \to S$. A set that is not countable Is uncountable.

Theorem 2.6. $\mathbb{{'}Z{}'}$ is countably infinite.

Proof. The function $f : \mathbb{{'}N{}'} \to \mathbb{{'}Z{}'}$ defined by

$f(n) = \begin{cases} n/2 & \mathrm{if}\; n\; \mathrm{is}\; even \\ -(n+1)/2 & \mathrm{if}\; n\; \mathrm{is}\; odd \end{cases}$

Is a bijection, enumerating $0, -1, 1, -2, 2, -3, 3, \ldots$ $\blacksquare$

Theorem 2.7. $\mathbb{{'}Q{}'}$ is countably infinite.

Proof. Every positive rational can be written as $p/q$ with $p, q \in \mathbb{{'}N{}'}^+$. Arrange the Pairs $(p, q)$ in an infinite grid and traverse them diagonally:

$1/1,\; 1/2,\; 2/1,\; 3/1,\; 1/3,\; 1/4,\; 2/3,\; 3/2,\; 4/1, \ldots$

Skipping duplicates (where $p/q = p'/q'$ in reduced form) yields an enumeration of $\mathbb{{'}Q{}'}^+$. Extending with negatives and zero gives an enumeration of $\mathbb{{'}Q{}'}$. $\blacksquare$

Theorem 2.8 (Cantor, 1891). $\mathbb{{'}R{}'}$ is uncountable.

Proof (Diagonal argument). Suppose for contradiction that $\mathbb{{'}R{}'}$ is countable. Then the Interval $[0, 1)$ can be listed as $r_1, r_2, r_3, \ldots$ where each $r_i$ has a unique decimal Expansion $r_i = 0.d_{i1}d_{i2}d_{i3}\ldots$ with each $d_{ij} \in \\{0, 1, \ldots, 9\\}$ (choosing the expansion that does not end in all 9s to avoid dual representations).

Define $s = 0.s_1 s_2 s_3 \ldots$ by

$s_i = \begin{cases} 5 & \mathrm{if}\; d_{ii} \neq 5 \\ 6 & \mathrm{if}\; d_{ii} = 5 \end{cases}$

Then $s \in [0, 1)$ and $s$ differs from $r_i$ in the $i$-th decimal place for every $i$ So $s \notin \\{r_1, r_2, \ldots\\}$Contradicting the assumption that the list was complete. Therefore $\mathbb{{'}R{}'}$ is uncountable. $\blacksquare$