Discrete Math for Data Science
DSCI 220, 2025 W1
September 5, 2025
Syllabus Continued
Communications
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|---|---|---|
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Logic
Truth
Make at least 3 observations about the following table:
| \(p\) | \(q\) | \(p \lor q\) |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
Summary
Key observations about the table:
| \(p\) | \(q\) | \(p \lor q\) |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
- The diagram is called a Truth Table
- p and q are variables, each of which take on one of two Boolean values, True or False (T/F, 1/0)
- ∨ is a binary operator that implements a function (creates output) from the 2 input variables to another Boolean value. It is characterized by its output values.
- The operator ∨ corresponds to the English word or.
- The expression p ∨ q is called a proposition.
Foreshadowing
| \(p\) | \(q\) | \(p \lor q\) |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
How many rows for a proposition on 3 variables? \(k\) variables?
Give a good name for an operator that takes 1 variable: __________
Give a good name for an operator that takes 3 variables: __________
How many different binary operators could there be?
06:00
Truth Tables
All graded learning activities in the course will use a tool called PrairieLearn.
Navigate to today’s activity and complete question 1
https://us.prairielearn.com/pl/course_instance/186238/
10:00
Beyond Operators
Propositions can be more complex than just a single operator!
Example:
| \(p\) | \(q\) | _____ | \(\neg(p \lor q)\) |
|---|---|---|---|
| F | F | T | |
| F | T | F | |
| T | F | F | |
| T | T | F |
WFFs
A well-formed formula is a Boolean statement generated by the following rules:
This definition is in the form of a Grammar.
_____________: <wff> and <atom>
_____________: p, q, r, s …
It is our first self-referential or recursive definition.
WFFs
Example derivation:
<wff>
WFFs (Notes)
Grammars can be used to construct many different kinds of sequences.
We could have included additional operators \(\rightarrow\), \(\leftrightarrow\), \(\oplus\), \(\uparrow\)
Computational evaluation of
<wff>is covered in DSCI221. For now, we trust Python and focus on logic.The
<wff>are propositions.
WFF Puzzle
Which of these are WFFs?
- \((\neg p ∧ (q ∨ r))\)
- \(((p ∧ q) ∨ (r ∧ \neg s))\)
- \(((\neg p) ∨ (q ∨ r))\)
- \((p ∨ (qr))\)
- \(\neg (p ∨ (q ∧ \neg r))\)
- \(\neg (p ∨ q))\)
03:00
WFFs
Theorem: \(\neg (p ∨ q))\) is not a WFF.
Proof:
Propositions
A proposition is a statement that can be either True or False.
Examples:
\(37 > 12\)
Fewer than 5 people in this room feel sleepy.
There are extra-terrestrial life forms.
This statement is False.
\(\underline{\hspace{10em}}\)
Translation
\(p\): ___________ ate cereal for breakfast.
\(q\): ___________ brought a backpack to class.
\((p\lor q)\)
\((p\wedge q)\)
\(\neg q\)
Predicates
\(p\): ___________ ate cereal for breakfast.
We may want to apply the statement to many students, in which case we define a Predicate.
\(P(x)\): \(x\) ate cereal for breakfast.
\(x\) can be instantiated to be a particular student, or an arbitrary student.
\(P(\underline{\hspace{2em}} )\) is a proposition.
Logical Equivalence
Ex: Is it true that \((p \lor q) \equiv (q\lor p)\) ?
Discussion points:
\(\equiv\) means logically equivalent
The answer had better be _________!!!
How can we justify our instinct?
Logical Equivalence example
Boolean Masks
Select all the cereals with at least 4 units of protein and no more than 6 units of sugars.
Words and Ideas
Truth Table
Proposition
Grammar
Predicate
Boolean Mask
Logical Equivalence