Discrete Math for Data Science

DSCI 220, 2025 W1

October 20, 2025

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Sets, Power Sets, & Cross Products

Goals

• Explore the Empty Set.
• Distinguish element (\(\in\)) vs subset (\(\subseteq\)).
• Define the power set \(\mathcal{P}(S)\) and its cardinality.
• Define Cartesian product \(A\times B\).

The Empty Set (\(\varnothing\))

Definition: \(\varnothing\) (a.k.a. \(\{\}\)) is the unique set with no elements.

• Cardinality: \(|\varnothing|=0\).
• Universal subset fact: For any set \(A\), \(\varnothing\subseteq A\).
• But membership is different: \(\varnothing\in A\) only if \(A\) literally contains the empty set as an element.

Examples

• If \(S=\{a,b\}\): \(\varnothing\subseteq S\), but NOT \(\varnothing\in S\).
  
• If \(T=\{\varnothing,\{a\}\}\): \(\varnothing\in T\).

Operations with \(\varnothing\)

For any set \(A\):

• Union: \(A\cup\varnothing = A\)
• Intersection: \(A\cap\varnothing = \varnothing\)
• Difference: \(A - \varnothing = A\), \(\ \varnothing\setminus A = \varnothing\)
• Complement (in \(U\)): \(U - \varnothing = U\)
• Cardinalities: \(|A\cup\varnothing|=|A|\), \(|A\cap\varnothing|=0\)

Quick demo with a tiny set on the board or in code; emphasize “subset vs element.”

DataFrame/Mask view of \(\varnothing\)

Think: empty mask = all False (selects no rows).

Element vs Subset

• \(x\in S\) means \(x\) is one item in \(S\).
• \(A\subseteq S\) means every element of \(A\) is in \(S\).

Examples (watch the braces!)

• If \(S=\{a,b\}\) then:
  • \(a\in S\) is true; \(\{a\}\in S\) is false; \(\{a\}\subseteq S\) is true.
  • \(\emptyset \in S\) is false; \(\emptyset \subseteq S\) is true.
• If \(T=\{\emptyset,\{a\}\}\) then:
  • \(\emptyset \in T\) is true; \(\{\emptyset\}\subseteq T\) is true.

Say aloud: “Curly braces change the type of thing. Elements vs sets of elements.”

Quick Checks

Let \(S=\{\texttt{tea},\texttt{coffee},\texttt{juice}\}\).

1. \(\texttt{tea}\in S\) ____
   
2. \(\{\texttt{tea}\}\in S\) ____
   
3. \(\{\texttt{tea}\}\subseteq S\) ____
   
4. \(\emptyset \subseteq S\) ____
   
5. \(\{\texttt{water}\}\subseteq S\) ____

Answers: 1✓, 2✗, 3✓, 4✓, 5✗.

Power Set

Definition: \(\mathcal{P}(S)\) = the set of all subsets of \(S\).

Example:

If \(S=\{a,b\}\), \(\ \mathcal{P}(S)=\) ________________________

 

Cardinality:

If \(|S|=n\), then \(|\mathcal{P}(S)|=\) ________________________

Why \(2^n\)?

Bitstring encoding:
Consider \(S=\{s_1,\dots,s_n\}\). For each subset \(T\subseteq S\) define an indicator of membership: \[(b_1,\dots,b_n)\ \text{where } b_i=\begin{cases}1,& s_i\in T\\ 0,& s_i\notin T\end{cases}.\]

How many subsets of a set of size \(n\)?

Connect to yesterday’s “\(b\) bits \(\Rightarrow 2^b\) codes.” Subsets \(\leftrightarrow\) \(n\)-bit strings.

Quick Check:

Let \(S=\{A,B,C\}\).

• Bitstring \(101\) corresponds to subset \(\{\_\_\_\_\}\).
• Subset \(\{B\}\) corresponds to bitstring \(\_\_\_\_\_\_\).

Empty Sets and Power Sets

• \(\varnothing\in \mathcal{P}(A)\) (since \(\varnothing\subseteq A\))
  
• \(\mathcal{P}(\varnothing)=\{\varnothing\}\) and \(|\mathcal{P}(\varnothing)|=1\)

Cartesian Product \(A\times B\)

Definition: \(A\times B=\{(a,b)\mid a\in A,\ b\in B\}\)

Cardinality: \(|A\times B|=|A|\cdot|B|\).

Example: \(A=\{\texttt{iced},\texttt{hot}\}\), \(B=\{\texttt{Latte},\texttt{Tea},\texttt{Mocha}\}\).

 

Then \[A\times B= \{(\texttt{iced},\texttt{Latte}),(\texttt{iced},\texttt{Tea}),(\texttt{iced},\texttt{Mocha}),\] \[(\texttt{hot},\texttt{Latte}),(\texttt{hot},\texttt{Tea}),(\texttt{hot},\texttt{Mocha})\},\]

and \(|A\times B|=2\cdot 3=6\).

Code