MATH1061/7861, Thu 17 Sep 2020
Assignment 1 marks: looking into this
Midsemester exam: during Monday lecture time on blackboard, under Assessments.
Will post an announcement over weekend,
9am-9.50am Brisbane time
Next lecture on Wednesday 9am.
Covers everything up to and including cardinalities and countability (Monday). Excludes relations. Automatically graded: multiple choice, true or false, numerical answer; NOT writing proofs
All open book, no human assistance!
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Question 2. Let 𝜌 and 𝜏 be two relations defined on R as follows:
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Question 3. Let R be the relation on Z defined by mRn if and only if 12 | m – n or 12 | m + n. Prove that R is an equivalence relation, and describe its equivalence classes.
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Question 4. Let n,m ∈ N, and let 𝜌 and 𝜏 be two relations defined on N as follows: ◦ a𝜌b ↔ a ≡ b (mod n);
◦ a𝜏b ↔ a ≡ b (mod m).
Let 𝜎 be the intersection 𝜎 = 𝜌 ∩ 𝜏.
Prove that 𝜎 is an equivalence relation, and describe the equivalence class [0] under 𝜎.
Question 5. Show that N × N × N is countable.
Challenge. Let S be the set (N × N × N × …), with a countably infinite number of N factors. Is S countable?