Boolean Logic: Propositional Calculus Recap
Attendance and logistics (classroom management)
Attendance policy emphasized: you must have been in class at least once by tomorrow to avoid being marked as a no-show.
Manual attendance noted due to no-shows; professor asks for students to confirm presence by tomorrow.
If a student cannot check-in due to a system issue, they should tell the instructor in class or send a direct message on Discord so it can be recorded later.
Emphasis on the impact of attendance on final grade; excessive unexcused absences (over three) will hurt the grade; medical trips, official team or band travel with documentation are excused up to three unexcused absences.
If on a project team, absences affect the team; repeated absences can lead to being kicked off a team.
Students must attend Friday sessions (R3 or R4 recitations); roster issues earlier in the term are now resolved.
Reminder: check Blackboard daily for tutoring announcements and course updates; Blackboard announcements can be unreliable at times (e.g., email notices may not reflect the latest PDF or link).
Plan to move communications to Discord for announcements, with Blackboard used for official items when necessary.
The instructor posted an announcement with a PDF about tutoring; students should access it via Blackboard (not directly from email).
Tutoring details: multiple tutors with specific time slots; one online evening option; QR code to schedule; regular tutoring in Library Room 206; thirty-minute or possibly hourly slots; slots fill up quickly; come prepared so tutors can help rather than do the work for you; tutoring is optional but encouraged to use peers’ help.
Weekly hours requirement: for this course, 2 hours per week; Elements course requires 2 hours as well; these hours do not overlap; can be scheduled in evenings if needed.
The instructor plans to add a dedicated Discord channel for rosters and communications.
The class will return to logic topics after these announcements.
Recap and goals: propositional calculus basics
Propositional calculus studies objects that can be true or false; evaluate statements for truth value.
Boolean algebra is a formal framework operating on true/false (1/0) values using logical operators: conjunction (AND), disjunction (OR), negation (NOT), and XOR, with implications (IF-THEN) also discussed.
Truth tables are the fundamental tool to enumerate all possible truth values of propositions and their compound forms; practice with truth tables builds intuition and speed, useful for exams.
The teacher emphasizes fluency with truth tables; aim is to be able to complete tables quickly, though some topics may require deeper understanding.
Quick recap of basic operators
Conjunction (AND): P \land Q
Intuition: both statements must be true.
Truth table intuition: true only when both P and Q are true.
Disjunction (OR): P \lor Q
Intuition: at least one statement is true; inclusive OR.
Truth table intuition: true if any operand is true (including when both are true).
Negation (NOT): \neg P
Intuition: flip the truth value.
Applies to a single proposition.
Exclusive OR (XOR): P \oplus Q or P Q
Intuition: exactly one of P or Q is true (one true and the other false).
Notation can vary; symbol often a plus inside a circle or ⊕.
Implication (IF-THEN): P \rightarrow Q
Intuition: if P is true, then Q must be true; P being false places no constraint on Q for the statement to be considered true.
Premise: P; Conclusion: Q; the rule connecting them is the implication.
Equivalence (not covered deeply in this session, but related): often written as P \leftrightarrow Q (not the focus here).
Truth tables (recap with explicit forms)
Conjunction: P \land Q
Rows: $(P,Q) = (T,T)
ightarrow T$, $(T,F)
ightarrow F$, $(F,T)
ightarrow F$, $(F,F)
ightarrow F$.
Disjunction: P \lor Q
Rows: $(T,T)
ightarrow T$, $(T,F)
ightarrow T$, $(F,T)
ightarrow T$, $(F,F)
ightarrow F$.
Negation: \neg P
Rows: $P=T
ightarrow\neg P=F$, $P=F
ightarrow\neg P=T$.
Exclusive OR: P \oplus Q
Rows: $(T,T)
ightarrow F$, $(T,F)
ightarrow T$, $(F,T)
ightarrow T$, $(F,F)
ightarrow F$.
Implication: P \rightarrow Q
Rows: $(T,T)
ightarrow T$, $(T,F)
ightarrow F$, $(F,T)
ightarrow T$, $(F,F)
ightarrow T$.
Negation: examples and caveats
Example: peanuts in peanut brittle (P) vs not having peanuts (¬P)
If P = true (peanuts present), then ¬P = false.
If P is true for peanuts in a product and you negate it, you flip the truth value.
Double negation: \neg(\neg P) \equiv P; beware of natural-language double negations (e.g., "you don't know nothing") which can be ambiguous.
Real-world illustration: XOR and ordinary OR
Ordinary OR in real life can often be inclusive (you can have both options).
XOR (exclusive OR) is when exactly one option is chosen or true; you cannot have both in the scenario.
Examples discussed in the class:
Salt and pepper for seasoning: often treated as regular OR (you can add salt, or pepper, or both).
Salt and sugar in recipes: more often exclusive (sometimes only salt or only sugar is appropriate; sometimes both depending on the recipe).
Vehicles example (car vs motorcycle): you can own a car, or a motorcycle, or both; XOR would capture cases like you need to fulfill one requirement but not both.
Early or late (not on time): you can be early or late; you cannot be both at the same time; if neither, you are on time (another interpretation of the scenario).
Important: the XOR truth table and the logical equivalences help model exclusive situations in software and data rules.
XOR in recitation (R3 vs R4) and logical equivalences
In the course, R3 and R4 sections can be modeled with XOR:
Truth: you attend R3 but not R4, OR you attend R4 but not R3.
Logical form: (R3 \land \neg R4) \;\lor\; (\neg R3 \land R4)
This is logically equivalent to the exclusive OR: R3 \oplus R4.
An alternative representation uses De Morgan-style form: (R3 \lor R4) \land \neg(R3 \land R4)
The instructor notes that such equivalences can be a useful way to convert between natural-language conditions and formal logic.
Implications (If-Then) in detail
Structure: P \rightarrow Q where P is the premise and Q is the conclusion.
How to evaluate: build a truth table with two sub-statements (P and Q) to see all combinations.
Example used in class: sprinkler on (P) implies grass wet (Q).
Premise: Sprinkler is on.
Conclusion: Grass is wet.
If sprinkler is on and grass is wet, the implication is true (no contradiction).
If sprinkler is off but grass is wet, this does not necessarily falsify the implication (could be due to other causes); implication can still be true because the premise did not occur.
If sprinkler is on but grass is not wet, the implication is false (the premise occurred but the conclusion did not).
If sprinkler is off and grass is not wet, the implication remains true (no evidence that the premise caused the conclusion).
Practical interpretation: in software or data analytics, an implication represents a rule that should hold given the premises; the only clear counterexample is the case P is true and Q is false.
Note on real-world interpretation: sometimes you may have additional factors (external failure, wrong wiring, etc.) that invalidate the rule; such cases require system diagnostics.
A helpful analogy the instructor used: “innocent until proven guilty” – an implication is not proven false unless you have a direct counterexample; otherwise you treat it as possibly true.
Premise and conclusion terminology
Premise: the initial condition or antecedent (P).
Conclusion: the result or consequent (Q).
In the sprinkler example: P = “the sprinkler is on,” Q = “the grass is wet.”
Truth-table analysis yields the four possible combinations of P and Q.
How to use these concepts in exams and coding practice
You will encounter truth-table fill-ins; practice by enumerating all possible truth values for multiple propositions.
Remember the semantics for each operator; misclassifying OR as XOR or vice versa is a common pitfall.
When translating natural-language statements into logic, identify conjunctions, disjunctions, negations, and implications; consider whether the OR is inclusive or exclusive.
For XOR, a convenient identity is: P \oplus Q \equiv (P \lor Q) \land \neg(P \land Q).
Practical study tips and course logistics (recap)
Regularly complete truth tables for all operators to build fluency; speed improves with practice.
Attend Fridays for recitations (R3 or R4) to reinforce understanding and teamwork.
If you have scheduling concerns for tutoring, use the QR code to select a course and time slot; tutoring slots are limited and fill up quickly; sessions are typically 30 minutes.
Tutoring hours do not overlap with your other course hours; plan ahead to fit both courses (two hours per course per week, non-overlapping).
The course emphasizes using Discord for announcements and Blackboard for essential items; stay on top of both platforms.
Install and use Anaconda with Spyder (as mentioned by the instructor) for practical coding practice; a reminder to ensure Python-based tools are ready in advance.
Summary of key formulas (LaTeX)
Conjunction: P \land Q
Disjunction: P \lor Q
Negation: \neg P
Exclusive OR: P \oplus Q
Implication: P \rightarrow Q
Equivalence (related): P \leftrightarrow Q
XOR equivalence forms: P \oplus Q \equiv (P \lor Q) \land \neg(P \land Q) and P \oplus Q \equiv (P \land
eg Q) \lor (
eg P \land Q)
Final takeaway
Mastery of Boolean logic, truth tables, and the correct use of logical operators is foundational for coding and data-driven decision-making; the more you practice, the faster you will be at building and debugging logical rules in software.