Discrete Mathematics for BSAI

01 · Opening Scenario

You enter a campus study room. A lamp is controlled by a motion sensor. The posted rule says: “If motion is detected, the lamp turns on.” You walk in and see the lamp on. Can you conclude that motion was detected?

Pause before reading on. The lamp may have been switched on manually. Seeing the lamp on does not, by itself, tell us why it is on. But if the sensor detected motion and the posted rule is operating as stated, then the lamp must be on. This difference between a rule and what can be concluded from an observation is the start of mathematical reasoning.

02 · Why This Matters

Mathematical reasoning asks: What do we know? What follows from it? What remains uncertain? Many systems make decisions from explicit conditions: a gate opens when an authorized card is present, a reminder is sent when an appointment is due, or an application displays a warning when two conditions occur together. Logic lets us describe and inspect such conditions precisely.

Later in this course we will prove statements, count possible cases, and study networks. Each task depends on distinguishing a statement from a question, interpreting the word “if,” and checking every possible truth-value case. These are skills you can learn without previous computing experience.

03 · Learning Objectives

By the end of Week 1, you should be able to:

1. Explain mathematical reasoning as drawing justified conclusions from stated information.
2. Decide whether a sentence is a proposition, and explain why.
3. Represent statements with letters such as P and Q.
4. Interpret ¬P, P ∧ Q, P ∨ Q, and P → Q.
5. Complete a two-proposition truth table and explain the false row of an implication.
6. Compare a logical expression with a small Boolean experiment in Colab.

Three-hour class guide: about 50 minutes for the scenario, reasoning, and proposition sorting; 65 minutes for connectives and truth tables; 45 minutes for worked systems, activities, and a short Colab demonstration; 20 minutes for practice feedback and an exit check.

Core classroom material Opening scenario through the combined truth table, selected worked examples, the Logic Playground, and one Colab cell. Self-study / extension The three-input worked table, further practice, and the ungraded mini mission can be completed after class. Begin every experiment by predicting; test; then explain the result in words.

04 · Intuitive Introduction

Consider four sentences: (a) “The study room lamp is on.” (b) “Please turn on the lamp.” (c) “Is the lamp on?” (d) “The room has 27 chairs.” Sentences (a) and (d) can be true or false in a specified situation, even if we do not yet know which. Sentence (b) is a request; (c) is a question. A request and a question are not true or false in the required sense.

We can combine ordinary statements. If P means “motion is detected” and Q means “the lamp is on,” then “motion is detected and the lamp is on” needs both parts; “motion is detected or the lamp is on” needs at least one part. The conditional “if motion is detected, then the lamp is on” imposes a requirement only when motion is detected.

05 · Mathematical Concepts

Reasoning from stated information

Begin by writing the rule and the observation separately. In the opening scenario the rule is P → Q. If we also know P, we may expect Q. If we know only Q, the rule does not tell us whether P occurred. In Week 2, we will use facts and rules to decide what conclusions actually follow; for now, focus on careful interpretation.

Four basic ways to combine propositions

The cards above are a preview. Work through each short journey below before using the full table in Section 07.

When more than one connective appears, parentheses tell us what is grouped: (P ∧ Q) → R is a rule about both P and Q together. Do not silently change the grouping.

06 · Formal Definitions and Notation

A proposition is a declarative statement that, in a specified context, is either true or false. We may not know its truth value, but it still has one. The value of a compound proposition depends on the values of its components.

07 · Visual Explanation

Think of a truth table as a complete list of possible situations. With two propositions there are four input combinations. The columns below show how each connective behaves.

08 · Step-by-Step Worked Examples

Compare AND and OR: If either an authorized card or a staff override can admit a person, let S mean “staff override is active.” Then the alternative entry condition is A ∨ S; when both are true, the inclusive OR remains true. The two policies answer different questions.

Converse check: Q → P would say that every time the lamp is on, motion was detected. Log 3, with Q=T, P=F, refutes that converse while satisfying the original rule. Our question is always what the given rule actually says.

09 · Real-System Scenario Examples

Scheduling two conditions

Let R mean “a classroom is free” and I mean “the instructor is free.” A meeting can proceed only if R ∧ I. If either condition fails, the combined condition is false. This illustrates why “and” is stricter than “or.”

Two ways to receive a message

Let E mean “email delivery succeeds” and S mean “SMS delivery succeeds.” “At least one channel succeeds” is E ∨ S. It remains true if both succeed. If a contract says “exactly one,” that is a different requirement.

A condition is not its converse

“If the alarm test button is pressed, a sound plays” is T → S. “If a sound plays, the test button was pressed” is S → T. An actual alarm can sound for other reasons, so the second statement need not follow from the first.

10 · AI/Computing Connection

A rule-based decision system evaluates clearly stated conditions. For example, “If the user has requested an accessible route and an accessible route exists, then display it” can be written using propositions about the request, availability, and display. Logic helps us inspect which combinations of input conditions a rule addresses and where information is missing.

This is one simple kind of decision logic, not a description of how every AI system works. Data-driven systems can also contain Boolean checks and stated policies, but a truth table does not by itself make a program intelligent or ensure that its inputs are accurate. In Week 2 we will ask whether conclusions actually follow from multiple rules and facts.

This is a simple rule-based decision. Modern AI systems can use much more complex methods, while Boolean checks remain useful in software and decision systems around them. The rule still depends on accurate inputs and a clearly stated policy.

11 · Python / Google Colab Exploration

Mathematics first, Python second. Predict each result by hand. The table translates notation, not the meaning of the scenario.

Open a blank Google Colab notebook. A cell is a space where you can enter code. Type the first block and press its Run button. The words True and False are Python’s Boolean truth values. print(...) displays a result; the parentheses enclose what to display. The signs = assign a value to a name. Here the names p and q stand for our propositions.

p = True
q = False
print(p)
print(q)

Python writes logical conjunction as and, disjunction as or, and negation as not. Each line below is an independent experiment using the values already assigned.

print(not p)     # negation of p
print(p and q)   # conjunction: both
print(p or q)    # inclusive disjunction: at least one

For a first check of an implication, reuse known operations: P → Q has the same truth values as ¬P ∨ Q. This identity is used here to inspect a small table; its formal justification through logical equivalence comes in Week 3.

p = True
q = False
print((not p) or q)

12 · Logic Playground and Interactive Activities

Practice / ungraded Try an answer, read the feedback, and revise it. Predict a result before touching a control; test it; then explain why. In the classroom, use cards and paper first, then compare with the screen.

13 · Check Your Understanding

1. Can a proposition be false? Give an example.
2. When are P ∧ Q and P ∨ Q different?
3. What is the only false case of P → Q?
4. If the lamp is on, does P → Q alone prove that motion was detected?

14 · Common Mistakes

15 · Practice Problems

Do Problems 1–5 by hand. Problems 6–8 extend the ideas. Use Colab only after you can explain an answer.

1. Classify: “12 is divisible by 3”; “Find the nearest room”; “Is the server on?”; “The campus bus departed at 8:00 today.”
2. Let P mean “it is raining.” Write ¬P in words. Does ¬P mean “it will never rain”?
3. If P=T, Q=F, determine ¬P, P ∧ Q, P ∨ Q, and P → Q.
4. A notification is sent when an assignment is submitted and notifications are enabled. Choose two propositions and write the condition symbolically.
5. Write the complete truth table for ¬P ∧ Q. Show the intermediate ¬P column.
6. An entry gate rule says: “If an authorized card is scanned, the green light appears.” The green light appears. What can you conclude from this rule alone? Give a possible alternative cause.
7. Let P mean “a route exists,” Q mean “the device has power.” Translate “a route exists and the device has power” and “a route exists or the device has power.” Compare their truth values when P=F, Q=T.
8. Write one compound proposition about a classroom rule and explain what each letter represents. Identify one observation that would make your rule false.

16 · Week 1 Mini Mission: Design a Smart Rule

Self-study / ungraded Choose a simple smart classroom, library or parking gate, reminder application, notification system, study-room access rule, or a similarly familiar system. This is a small personal practice task, not a major semester case study.

Try a prediction first: Which row do you expect to be most informative? Test it; then explain what your rule does and does not say.

17 · Challenge Problem

A smart lock uses two rules: “If the card is valid, the indicator turns green” and “If the indicator is green, the door unlocks.” Let V, G, and U stand for these three statements. Translate both rules. If a log records V=T, G=T, U=F, identify which rule is violated. If the log only records U=T, can you infer that the card was valid?

18 · Summary

A proposition has a truth value even when that value is unknown to us. Negation reverses a truth value. Conjunction requires both inputs; inclusive disjunction requires at least one. An implication P → Q is false only when its premise is true and its conclusion false. Truth tables list all input combinations and expose invalid interpretations such as silently reversing a conditional.

In mathematical reasoning, state what is known, translate the rule accurately, check relevant cases, and explain what does or does not follow.

19 · Concept Map

Reflection: Which step would be missed if you wrote code before deciding what the real-world statements mean?

20 · Key Terms

21 · What Comes Next

Next week, statements will gain variables and quantifiers such as “every” and “some.” You will combine facts and rules in the major case study Can the System Conclude This? For now, choose one rule you encounter in everyday life. Write its premise and conclusion, find the observation that would violate it, and ask whether its converse must be true.

Exit reflection: “I once thought an if–then statement meant ______. Now I understand ______.”