01 · Assignment Overview
A short reasoning portfolio: evaluate rules, decide what follows from stated facts, and write concise proofs. Complete after Week 4.
Weeks 1–4
25
75–90 minutes
Learning evidence: Truth table and equivalence, finite-domain inference, a direct and a contrapositive proof, and a complete induction step.
Instructions: Show the mathematical steps and explain each conclusion in one or two sentences. State any assumption you use. Hand reasoning and diagrams are required; Python or Colab may check a prediction but cannot replace the argument. Write your name and roll number on the first page.
Submission: One readable PDF of your own written or typed work, with pages in question order. Include diagrams and tables where requested. The instructor will announce the date and course-grade weight separately; the 25 points below describe only this assignment's internal rubric.
02 · Questions
Q1 · Two forms of a rule 6 points
Let P mean “the student completed the checkpoint” and Q mean “the student may open the next activity.” A system policy says P → Q.
- Construct the four-row truth table for P→Q and ¬Q→¬P. (4)
- Are these statements equivalent? State one row that shows why the converse Q→P cannot be inferred from the policy. (2)
Q2 · Facts, rules and three verdicts 7 points
The domain is D={Amina, Bilal, Huda}. M(x) means “x missed a checkpoint”, H(x) “x requested help”, I(x) “invite x” and N(x) “notify an advisor about x”. The only recorded facts are M(Amina), H(Amina), M(Bilal), and ¬H(Huda). A missing fact is unknown.
Rules for every learner in D: M(x)∧H(x)→I(x); I(x)→N(x); ¬H(x)→¬I(x).
- Write the first rule with ∀ and its stated domain. (1)
- For each proposal, give Follows, Contradicted or Not enough information and one reason: I(Amina), N(Amina), I(Bilal), I(Huda). (4)
- Give the two-step chain that establishes N(Amina). Explain why the record for Bilal does not establish ¬I(Bilal). (2)
Q3 · Two short proofs 8 points
- Prove directly: the sum of any two even integers is even. Name the arbitrary integers and justify the final even form. (4)
- Prove by contraposition: if n² is even, then n is even, for every integer n. Begin with “If n is odd…” and show why n² is odd. (4)
Q4 · One induction proof 4 points
Prove for all integers n≥1 that 1+2+⋯+n=n(n+1)/2. Identify the base case, inductive hypothesis and algebraic step from n=k to n=k+1. Checking three examples is not a substitute for the step.
03 · Submission Checklist
- Four truth-table rows with both columns, plus a converse counter-row.
- Four three-way verdicts supported by the exact facts and rules.
- Proofs begin with arbitrary integers, not only numerical examples.
- Induction has a true first case and a justified k→k+1 calculation.
Mark distribution: Q1 6 + Q2 7 + Q3 8 + Q4 4 = 25. Optional code checks receive no separate points.