The Timpson Lyceum · Discrete Math (2025)

Factorial, Sets, and Quantifiers - Lecture 5 (Discrete Math)

Factorial, Sets, and Quantifiers – Lecture 5

Overview

This lecture pulls together the counting tools from the previous week and builds the foundational language of set theory and logic. It introduces factorials and falling factorials as ways to count arrangements, then compact notation for long sums and products (sigma and pi). From there it develops sets — membership, equality, cardinality, subsets, set-builder notation, and the power set — and practices writing rigorous proofs of set equality. It closes with quantifiers (existential and universal), how to read nested quantifier statements, and the mechanical rules for negating them.

Key Concepts

Counting choices (with vs. without repeats)

• Making $k$ choices from $n$ options with repeats: $n^k$.
• Making $k$ choices from $n$ options without repeats: the falling factorial $n_{(k)} = n(n-1)(n-2)\cdots(n-k+1)$ — multiply $k$ terms, decrementing by one each time.
• On a test, leave such an answer as the unevaluated product (e.g. $18 \cdot 17 \cdot 16 \cdot 15$). The product shows your reasoning; a single huge number hides any arithmetic slip.

Factorial

• $n!$ ("$n$ factorial") $= n(n-1)(n-2)\cdots 2 \cdot 1$ for a natural number $n$. It equals the falling factorial $n_{(n)}$.
• Example: $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$.
• Interpretation: $n!$ counts the number of ways to arrange $n$ distinct objects in an ordered list.
• $0! = 1$: there is exactly one way to arrange zero objects (the empty arrangement), not zero ways.

Summation notation (sigma)

• $\sum_{k=1}^{n} a_k$ means start the dummy variable $k$ at the lower bound, substitute each integer up to the upper bound $n$, and add all the results.
• Example: $\sum_{k=2}^{7}(2k+1) = 5 + 7 + 9 + 11 + 13 + 15$.
• The notation lets you write very long sums compactly, e.g. $\sum_{k=1}^{500} 2k$ is all even integers from $2$ to $1000$.

Product notation (pi)

• $\prod_{k=1}^{n} a_k$ works exactly like a sum but multiplies the terms. Capital pi replaces capital sigma.
• $\prod_{k=1}^{n} k = n!$.
• Empty product / empty sum: $\prod_{k=1}^{0} k = 1$ and an empty sum $= 0$. These are the multiplicative and additive identities — splitting a big product/sum into groups, a group with no terms must contribute the identity so the total is unchanged.

Sets

• A set is an unordered, unique collection: order doesn't matter and repeated elements don't matter.
• Notation: list elements separated by commas inside curly braces, e.g. $\{1, 2, \text{SpongeBob}, \{2,3\}\}$ — this set has four elements ($\{2,3\}$ counts as a single object).
• $\{2, 2, 3\}$ has only two elements; duplicates collapse.
• Membership: $x \in A$ means $A$ is a set and $x$ is one of its elements; $x \notin A$ means it is not. Caution: $1$ and $\{1\}$ are different objects, so $1 \notin \{\{1\}, 2\}$ while $\{1\} \in \{\{1\}, 2\}$.

Set equality, cardinality, finiteness

• Equality: $A = B$ if and only if every element of $A$ is in $B$ and every element of $B$ is in $A$.
• Cardinality $|A|$ is the number of distinct elements in $A$. E.g. $|\{1, 3, 2, 2, 1, 1\}| = 3$.
• A set is finite if its cardinality is an integer; otherwise it is infinite. The naturals $\mathbb{N} = \{0, 1, 2, 3, \dots\}$ are infinite ($|\mathbb{N}| = \infty$, and $\infty$ is not an integer).
• $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ is the integers; $\mathbb{N}$ is the natural numbers.

Set-builder notation

• Form: $\{\, x : P(x) \,\}$ — a dummy variable, a colon read as "such that," then a predicate (a condition that is true or false). $x$ is in the set exactly when $P(x)$ is true.
• Evens: $\{\, x : x \in \mathbb{Z} \ \text{and}\ 2 \mid x \,\}$, often shortened to $\{\, x \in \mathbb{Z} : 2 \mid x \,\}$. (List membership in the base set first so the divisibility condition makes sense.)
• Odds: $\{\, n \in \mathbb{Z} : \exists\, m \in \mathbb{Z},\ n = 2m + 1 \,\}$.
• Perfect squares: $\{\, y \in \mathbb{Z} : \exists\, x \in \mathbb{Z},\ y = x^2 \,\}$.

Subsets and the power set

• $A \subseteq B$ ("subset") means every element of $A$ is also in $B$. The symbol has a line underneath, allowing equality; a proper subset $\subset$ forbids equality.
• Every set is a subset of itself: $A \subseteq A$.
• Don't confuse $\in$ with $\subseteq$: $\{1,2\} \subseteq \{1,2,3\}$ is true, but $\{1,2\} \subseteq \{\{1,2\}, 3\}$ is false (its elements $1, 2$ are not in the right-hand set).
• The empty set $\varnothing = \{\}$ is the set with no elements.
• Counting subsets: a set with $n$ elements has $2^n$ subsets. For each element you make an independent yes/no choice of whether to include it.
• The power set $\mathcal{P}(A)$ is the set of all subsets of $A$, so $|\mathcal{P}(A)| = 2^{|A|}$.

Quantifiers

• Existential $\exists$ ("there exists" / "some"), always followed by "such that": $\exists\, n \in \mathbb{N},\ n \text{ is odd}$.
• Universal $\forall$ ("for all" / "for every"): $\forall\, x \in \mathbb{N},\ x \ge 0$.
• Repeated same-set quantifiers may be combined: $\forall x \in \mathbb{Z}, \forall y \in \mathbb{Z}$ becomes $\forall x, y \in \mathbb{Z}$.
• Order matters. $\exists x \in \mathbb{Z}, \forall y \in \mathbb{Z},\ x + y = 0$ is false (no single $x$ works for every $y$), but $\exists x, y \in \mathbb{Z},\ x + y = 0$ is true.
• Negation rules — push the negation inward and flip the quantifier:

"Not all swans are white" becomes "there exists a swan that is not white."

Worked Example

Claim: The set of even integers $E = \{\, x \in \mathbb{Z} : 2 \mid x \,\}$ equals the set of integers expressible as a sum of two odds, $F = \{\, z \in \mathbb{Z} : z = a + b,\ a, b \text{ odd} \,\}$.

Prove set equality by showing both inclusions: $E \subseteq F$ and $F \subseteq E$.

($E \subseteq F$) Take any $n \in E$. Being even means $n = 2m$ for some integer $m$. Rewrite it as a sum of two odds:

Both summands are odd, so $n \in F$. (Concretely, $8 = 2\cdot 4 = (2\cdot 4 + 1) + (-1) = 9 + (-1)$.)

($F \subseteq E$) Take any $n \in F$, so $n = a + b$ with $a, b$ odd. Then $a = 2x + 1$ and $b = 2y + 1$ for integers $x, y$:

Since $x + y + 1$ is an integer, $n$ is two times an integer, hence even, so $n \in E$. (Concretely, $18 = 15 + 3 = (2\cdot 7 + 1) + (2\cdot 1 + 1) = 2(7 + 1 + 1) = 2\cdot 9$.)

Both inclusions hold, therefore $E = F$. $\blacksquare$

Summary

• Count $k$ choices from $n$ options as $n^k$ with repeats and as the falling factorial $n(n-1)\cdots(n-k+1)$ without; $n!$ counts the arrangements of $n$ objects, and $0! = 1$.
• $\sum$ (sigma) and $\prod$ (pi) compactly write long sums and products; the empty sum is $0$ and the empty product is $1$.
• A set is an unordered collection of unique elements; equality requires mutual inclusion, $|A|$ is its element count, and set-builder notation $\{x : P(x)\}$ describes infinite sets via a predicate.
• A set of $n$ elements has $2^n$ subsets, collected in the power set $\mathcal{P}(A)$ with $|\mathcal{P}(A)| = 2^{|A|}$.
• Prove two sets equal by showing each is a subset of the other, supplying a general procedure that works for any chosen element.
• Read $\exists$ as "there exists … such that" and $\forall$ as "for all"; quantifier order changes meaning, and negating flips $\forall \leftrightarrow \exists$ while negating the inner predicate.