Discrete Mathematics for BSAI

01 · Opening Scenario

A study-support system knows that Ayesha missed the quiz. Its rule says: “Every learner who missed the quiz should be offered a review meeting.” The screen proposes: Offer Ayesha a review meeting.

Can the system conclude this? Yes. Ayesha satisfies the condition in the rule. Now suppose the system proposes: Offer Sara a review meeting. We have said nothing about Sara. Her case may qualify under this or another rule, but the information shown does not establish it.

02 · Why This Matters

A rule about “every learner” is broader than a rule about one named learner. To use it, we must say which people are under discussion and check whether the known facts meet its condition. This is useful in eligibility checks, reminders, access rules, and transparent support systems.

Central question: Given some facts and rules, what can the system actually conclude? Logical reasoning can show a decision follows from a stated model. It cannot guarantee the recorded facts are accurate or that the policy itself is fair. Those are separate questions.

03 · Learning Objectives and Class Guide

1. Explain a predicate, variable, and domain in ordinary words and notation.
2. Read, translate, and negate simple “every” and “at least one” claims.
3. Use explicit facts and rules to justify a conclusion, reject a contradicted claim, or identify insufficient information.
4. Trace a short chain of two rules without reversing either rule.
5. Predict a decision by hand, then check a small Boolean Python example.

Three-hour class guide: 35 minutes for scenario and predicates; 40 for quantifiers and negations; 40 for facts, rules, and Inference Detective; 45 for the case study; 20 for a brief Colab demonstration, feedback, and exit check.

Classroom essentials Sections 01–10 and one Colab cell. Self-study Remaining practice and selected solutions. Detective guesses, discussion, and Python experiments are ungraded; a brief quiz or case response may be formally assessed by the instructor.

04 · Predicates: Statements with a Place to Fill

“Ayesha missed the quiz” can be true or false in a specified context. But “x missed the quiz” has a blank: we need to know who x is. We call M(x) a predicate or propositional function. It becomes a proposition when we replace x with a particular learner (or quantify it, as we will shortly).

A variable is a placeholder such as x. A domain is the set of allowed values for that placeholder. For example, let D = {Ayesha, Bilal, Sara} be the learners in a small tutorial group. Then M(Ayesha) means “Ayesha missed the quiz.” The same notation does not automatically tell us whether M(Sara) is true.

05 · Quantifiers: “Every” and “At Least One”

Ordinary language often talks about a whole group. The universal quantifier ∀ means “for every member of the stated domain.” The existential quantifier ∃ means “there is at least one member of the domain.” Say the domain aloud whenever you read a quantified expression.

In the last row, M(x) → C(x) is inside the scope of “for every.” Learners who did not miss the quiz are not required by this rule to need a check-in. The rule also does not say that only learners who missed a quiz can need a check-in.

06 · Negating “Every” and “At Least One”

Suppose someone says, “Every learner submitted.” To show this is false, you need one learner who did not submit. Suppose someone says, “At least one learner submitted.” To show that is false, you need to establish that none submitted.

Here ¬ means “not” and ≡ means the two expressions make the same claim. To negate the whole claim, swap “every” with “at least one” and negate the property. Keep the same domain. This week we use the pattern to read claims; Week 3 will examine equivalence more systematically.

07 · Facts, Rules, and What Follows

A fact is a stated item in our model, such as M(Ayesha). A rule connects conditions to a consequence, such as ∀x∈D, M(x) → C(x). We may use the rule for Ayesha because she is in D and M(Ayesha) is known. Thus C(Ayesha) follows.

We need three possible verdicts when checking a proposed conclusion P:

Distinguish the verdicts: A conclusion that does not follow is not necessarily false. Use Contradicted only when the opposite is established, and Not enough information when neither side is established.

Two-rule chain

Suppose every learner who misses a quiz needs a check-in, and every learner who needs a check-in should be offered a meeting. From M(Ayesha), first obtain C(Ayesha); then obtain O(Ayesha). Writing the middle step keeps the reasoning inspectable.

08 · Worked Examples: Understand → Represent → Reason

09 · Inference Detective

Ungraded exploration First predict a verdict without pressing a button. Then test it and explain the reasoning in your own words. “Contradicted” means the opposite is established; “Not enough information” means neither side is established.

10 · Major Case Study 1: Can the System Conclude This?

Scenario. A student-support system suggests review meetings. It is meant to be transparent: a staff member can see exactly which recorded fact and policy rule support a suggestion. It does not diagnose a learner or evaluate academic ability.

Understand · What is known?

The domain for this miniature case is D={Ayesha, Bilal, Sara}. The recorded positive facts are: Ayesha missed the quiz; Bilal requested help. No quiz-status or help-request fact is supplied for Sara. The rules are:

1. Every learner who missed the quiz needs a check-in.
2. Every learner who needs a check-in should be offered a review meeting.
3. Every learner who requested help should be offered a review meeting.

Represent · Build the mathematical model

Facts: M(Ayesha) and H(Bilal).

Rules: ∀x∈D(M(x) → C(x)); ∀x∈D(C(x) → O(x)); ∀x∈D(H(x) → O(x)).

Reason · Trace decisions step by step

Applying Rule 1 to Ayesha gives C(Ayesha). Applying Rule 2 to that new conclusion gives O(Ayesha). Separately, H(Bilal) and Rule 3 give O(Bilal). Therefore ∃x∈D, O(x) follows: Ayesha is already a witness (and Bilal is another).

Test · Predict, then inspect

Before looking back at the steps, classify these three proposed decisions: O(Ayesha); O(Bilal); O(Sara). State the first fact and last rule for each supported decision. Explain why no negative decision about Sara is justified.

Interpret · What does the mathematics say?

The model justifies offering meetings under a stated policy; it does not say that anyone attended one. The case illustrates how universal rules apply to named individuals, how two rules can be chained, how one witness supports an existential statement, and why absent data should not be turned into a negative claim. Discuss whether a staff review and accurate records are needed before the system acts.

Reflection: If the system later records H(Sara), which new conclusion follows? Would that prove Sara missed the quiz? Explain both answers.

11 · Python / Google Colab Exploration

Do the case-study reasoning on paper first. In a Colab cell, a variable stores a value. Week 1 introduced True, False, and print(). This week if checks a Boolean condition; the indented line runs if it is true. else supplies an alternative branch. A colon begins each branch. We test only one named learner; this code does not implement the universal quantifier or solve the whole case study.

missed_quiz = True

if missed_quiz:
    needs_check_in = True
    offer_meeting = needs_check_in
    print("Offer Ayesha a review meeting:", offer_meeting)
else:
    print("This rule alone does not decide about Ayesha.")

12 · Check Your Understanding and Practice

Try 1–5 independently. Questions 6–7 extend the reasoning. Use the selected solutions at the end only after recording a prediction and an explanation.

1. With domain D={Ali, Sana}, define P(x) as “x is present.” Is P(x) already a proposition? Is P(Ali) one?
2. Translate “Every registered student has an ID” and “Some registered student has an ID.” Define the predicates and domain before writing symbols.
3. Negate ∀x∈D, R(x) in symbols and ordinary words. Negate ∃x∈D, R(x).
4. Rule: every overdue book triggers a notice. Fact: Book A is overdue. Does “Book A triggers a notice” follow? Show the particular fact and rule.
5. Rule: every valid card opens the gate. Fact: the gate opened for Bilal. Does “Bilal had a valid card” follow? Give another possible explanation.
6. For the complete domain D={a,b,c}, P(a)=T, P(b)=T, P(c)=F. Evaluate ∀x∈D, P(x) and ∃x∈D, P(x). Identify a counterexample and a witness.
7. Facts: R(Nida). Rules: ∀x(R(x)→C(x)) and ∀x(C(x)→S(x)). Trace a two-step conclusion. Does S(Omar) follow if nothing is known about Omar?

13 · Common Mistakes

14 · Summary

A predicate P(x) becomes a statement about an individual when x is specified. A domain identifies which values x may take. ∀ means every domain member; ∃ means at least one. To negate “every,” exhibit one exception; to negate “some,” establish none. Apply rules only when their stated conditions are supported. A multi-step conclusion should show its intermediate step.

Reasoning habit: A proposed conclusion may be established, contradicted, or undetermined. Ask which of those descriptions is justified by the information actually given.

15 · Concept Map

Reflect: Which branch of the map prevents us from interpreting a missing record as false?

16 · Key Terms

17 · What Comes Next

In Week 3, we will ask why a reasoning pattern is valid across possible cases. We will connect implication with logical equivalence, then build short direct and indirect proofs. Keep this week’s central habit: write down exactly which conclusion follows from which facts and rules.

Exit reflection: “When a fact is missing, I should ______, because ______.”