The Timpson Lyceum · Computation Theory

Lecture 2 (Computation Theory) Set Theory and Functions

Set Theory and Functions – Lecture 2

Overview

This lecture builds set theory on top of logic and uses it as the foundation for the rest of the course — every mathematical object, including the objects of computation theory, ultimately reduces to a set. It introduces what a set is, the standard number sets, and set-builder notation, then develops the relations and operations that act on sets (subset, equality, union, intersection, difference, complement, and Cartesian product). From there it moves up the ladder from lists and tuples to relations, and finally to functions and their key properties: onto, one-to-one, and bijection. A recurring theme is learning to read dense symbolic notation precisely rather than relying on intuition or pictures.

Key Concepts

Set. An unordered, unique collection of objects, written between curly braces, e.g. $\{1, 2, \square\}$. "Unique" means repeated elements do not matter ($\{1,2,2\} = \{1,2\}$); "unordered" means order does not matter ($\{1,2\} = \{2,1\}$). Formally a set is an undefined term whose meaning comes from the axioms.

Membership. $x \in A$ means $x$ is an element of the set $A$; $x \notin A$ means it is not. Note that $\{2\} \in A$ and $2 \in A$ are different claims — the set $\{2\}$ is a distinct object from $2$.

Standard sets.

• $\varnothing$ — the empty set, the set with nothing in it.
• $\mathbb{N}$ — the natural numbers. In this course $\mathbb{N} = \{0, 1, 2, 3, \dots\}$ (the zero is included).
• $\mathbb{Z}$ — the integers, $\{\dots, -2, -1, 0, 1, 2, \dots\}$.
• $\mathbb{Q}$ — the rationals, any fraction of integers such as $\tfrac{3}{2}$ or $-\tfrac{15}{91}$.
• $\mathbb{R}$ — the reals. Numbers like $\pi$ and $\sqrt{2}$ are real but not rational (irrational).

The "$\dots$" convention. Three dots mean: continue the obvious pattern indefinitely. It signals an infinite set, not the literal symbol "infinity."

Set-builder notation. $\{\, x : P(x) \,\}$ — the set of all $x$ such that the predicate $P(x)$ is true. The colon reads "such that." A shorthand pulls the source set out front: $\{\, x \in \mathbb{Z} : x \ge 0 \,\}$ means the same as $\{\, x : x \in \mathbb{Z} \text{ and } x \ge 0 \,\}$. Inside the condition, a comma can be read as "and."

• Naturals: $\mathbb{N} = \{\, x \in \mathbb{Z} : x \ge 0 \,\}$.
• Even integers: $\{\, n : n = 2k,\ k \in \mathbb{Z} \,\}$.
• Perfect squares $\{0,1,4,9,\dots\}$: $\{\, n : n = k^2,\ k \in \mathbb{Z} \,\}$.

Set relations (produce a true/false value):

• Equality — $A = B$ iff every element of $A$ is in $B$ and every element of $B$ is in $A$: $(\forall x \in A,\ x \in B) \wedge (\forall x \in B,\ x \in A)$.
• Subset — $A \subseteq B$ iff $\forall x \in A,\ x \in B$. Behaves like "$\le$." Equivalently, $A = B$ iff $A \subseteq B$ and $B \subseteq A$.
• Proper subset — $A \subset B$ iff $A \subseteq B$ and $A \ne B$. Behaves like "$<$."
• Superset — $B \supseteq A$ is just $A \subseteq B$ read in reverse.

Lists and tuples. A list is an ordered collection, written with parentheses, e.g. $(1,2,2)$. For lists, order and duplicates matter, so $(1,2) \ne (2,1)$ and $(1,2,2)$ has three entries. An ordered pair is a two-element list. An $n$-tuple is a list with exactly $n$ elements.

Set operations (take sets, return a set):

• Union — $A \cup B = \{\, x : x \in A \text{ or } x \in B \,\}$. (The "or" of set operations.)
• Intersection — $A \cap B = \{\, x : x \in A \text{ and } x \in B \,\}$. (The "and" of set operations.)
• Difference — $A \setminus B = \{\, x : x \in A \text{ and } x \notin B \,\}$. Start with $A$, remove anything also in $B$.
• Complement — $\overline{A} = \{\, x : x \notin A \,\}$, taken relative to an understood universe $U$: everything in $U$ that is not in $A$.
• Cartesian (cross) product — $A \times B = \{\, (x,y) : x \in A,\ y \in B \,\}$, all ordered pairs with first coordinate from $A$ and second from $B$. Order matters: $A \times B \ne B \times A$ in general. If $|A| = m$ and $|B| = n$, then $|A \times B| = mn$. This generalizes to $A_1 \times A_2 \times \cdots \times A_n = \{\,(a_1, a_2, \dots, a_n) : a_i \in A_i\,\}$, and $A^2 = A \times A$.

Relation. A set whose every element is an ordered pair. $\{(1,2),(\square,\triangle),(3,4)\}$ is a relation; $\{(a,b), \{2,3\}\}$ is not (the second element is a set, not a pair). The empty set is vacuously a relation. If $R$ is a relation, we write $x\,R\,y$ to mean $(x,y) \in R$. Example: "$<$" is the relation $\{\,(x,y) : x < y\,\}$, so "$1 < 2$" just says $(1,2) \in\, <$. A set-theoretic relation is a more restrictive notion than a logical relation (a two-variable predicate), but it still behaves like one.

Function. A relation $f$ is a function iff whenever $(a,b) \in f$ and $(a,c) \in f$, then $b = c$. In words: every input has exactly one output. Equivalently, $a = b \implies f(a) = f(b)$. If $(a,b) \in f$ we write $f(a) = b$. Asking for $f(x)$ where $x$ is not an input, or applying $f(\cdot)$ notation to a non-function, is meaningless.

• Domain of $f$: $\{\, x : \exists y,\ (x,y) \in f \,\}$ — the set of all inputs (left-hand elements).
• Image (a.k.a. range): $\{\, y : \exists x,\ (x,y) \in f \,\}$ — the set of all outputs actually achieved.

Function-type notation. $f : A \to B$ means $f$ is a function with domain equal to $A$ and image a subset of $B$ (not necessarily all of $B$). Here $A$ and $B$ are sets and $f$ is a function. The full reading of $f(x) = x^2$ from algebra is "$f : \mathbb{R} \to \mathbb{R}$ by $f(x) = x^2$," shorthand for $\{\,(x,y) : x \in \mathbb{R},\ y = x^2\,\}$. A function is also called a map.

Properties of functions (for $f : A \to B$):

• Onto (surjective) — the image of $f$ equals $B$ (every element of $B$ is hit). There is no symbol for this; you state it.
• One-to-one (injective), written $1\text{-}1$ — whenever $(a,c) \in f$ and $(b,c) \in f$, then $a = b$. In words: every output comes from a unique input. Equivalently $f(a) = f(b) \implies a = b$, or contrapositively, different inputs give different outputs.
• Bijection — both one-to-one and onto. A bijection pairs each element of $A$ with one element of $B$ and vice versa. Two sets have the same size exactly when a bijection exists between them.

Reference examples.

• $f(x) = x^2$ on $\mathbb{R}$: not one-to-one (since $2^2 = (-2)^2$) and not onto $\mathbb{R}$ (negatives are never outputs; image is $[0, \infty)$).
• Absolute value $|\cdot| : \mathbb{Z} \to \mathbb{Z}$: not one-to-one ($|3| = |-3|$) and not onto (no input gives $-4$).
• $g(x) = 2x + 1$ on $\mathbb{R}$: both one-to-one and onto (every output traces back to a unique input).
• $\mathbb{Z}_m$ notation: $\mathbb{Z}_m = \{0, 1, 2, \dots, m-1\}$, the first $m$ whole numbers; e.g. $\mathbb{Z}_4 = \{0,1,2,3\}$.

Worked Example

Goal: decide whether $f : A \to B$ with $A = \{1,2,3\}$, $B = \{4,5,6\}$ and $f = \{(1,4),(2,5),(3,6)\}$ is a bijection. The criteria are: one-to-one means equal outputs force equal inputs, and onto means the image equals $B$.

By contrast, $g = \{(1,6),(2,6),(3,6)\}$ is a function with domain $A$, but its image is $\{6\} \subsetneq B$, so $g$ is not onto; and since $g(1) = g(2)$ with $1 \ne 2$, it is not one-to-one either. The relation $\{(1,4),(2,5),(1,6)\}$ fails earlier still — $1$ maps to both $4$ and $6$, so it is not even a function.

Summary

• A set is an unordered, unique collection; duplicates and order never matter. Lists (parentheses) are ordered and do count duplicates.
• Memorize the standard sets $\varnothing, \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ (here $\mathbb{N}$ includes $0$), and use set-builder notation $\{x : P(x)\}$ for precise, even infinite, descriptions.
• Relations on sets ($=$, $\subseteq$, $\subset$) return true/false; operations on sets ($\cup$, $\cap$, $\setminus$, $\overline{\,\cdot\,}$, $\times$) return new sets. For the cross product, $|A \times B| = |A|\,|B|$ and order matters.
• A relation is any set of ordered pairs; a function is a relation in which each input has exactly one output; $f : A \to B$ fixes the domain as $A$ and constrains the image to a subset of $B$.
• A function is onto when its image is all of $B$, one-to-one when each output has a unique input, and a bijection when it is both — and bijections are exactly what it means for two sets to have the same size.
• The hard part early on is the notation, not the ideas: practice reading dense symbolic statements precisely instead of leaning on pictures or graphs.