Equivalence relation practice question

AI Summary

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This lecture demonstrates how to verify whether a relation on the integers is an equivalence relation, then applies that idea to a multiple-choice question about ordered pairs. The instructor defines R1 on Z by (a,b) in R1 iff a+b is even, tests concrete pairs such as (-3,-3), (1,4) X, and (13,11) y, and organizes the proof into rows labeled Z, R, S, T. He shows reflexivity using -3 + (-3) = -6 and the pair (-3,-3), symmetry from a+b = b+a, and transitivity by assuming (a,b) in R and deriving the required condition. A second relation R2, defined by a+b odd, is introduced to contrast with R1 and fails the reflexivity test. The lesson ends with a question asking for the largest and smallest number of ordered pairs in equivalence relations on an n-element set S; option (B) n^2 and n is circled as correct.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The instructor introduces the relation R1 : (a, b) iff (a + b) is even over the set of integers. He underlines (a, b), writes Z beneath it to mark the underlying set, and draws a horizontal integer number line labeled -3 through +3 with arrows. He then begins testing the relation by writing -3 + (-3) = -6 and placing the ordered pair (-3,-3) beside it, showing a concrete instance where the sum is even.

2. 2:00 – 5:00 02:00-05:00

   The board is organized into rows labeled Z, R, S, and T to structure the equivalence-relation proof. The R row contains examples such as (-3,-3) and (1,4) X, while the S row shows if (a,b) in R then (b,a) in R and the equation a+b = b+a. The instructor writes ig (a,b) in R to begin a formal proof step, using specific integer pairs and the commutativity of addition to demonstrate reflexivity, symmetry, and transitivity for R1.

3. 5:00 – 6:29 05:00-06:29

   The instructor introduces R2 : (a, b) iff (a + b) is odd over the set of integers to contrast with R1, noting that an odd-sum relation fails reflexivity. The lesson then transitions to a multiple-choice question: Let S be a set of n elements; what are the largest and smallest numbers of ordered pairs in equivalence relations on S? Options (A) n and n, (B) n^2 and n, (C) n^2 and 0, and (D) n and 1 appear, and the instructor circles option (B) n^2 and n as the correct answer.

The central idea is that an equivalence relation must satisfy reflexivity, symmetry, and transitivity. For R1 on Z defined by even sums, the instructor verifies each property: reflexivity is shown because a+a = 2a is always even (illustrated by -3 + (-3) = -6), symmetry follows from a+b = b+a, and transitivity is argued by assuming (a,b) in R1 and deriving the needed condition. The contrasting relation R2, based on odd sums, fails reflexivity because a+a is never odd. The final multiple-choice question connects equivalence relations to ordered pairs: the largest equivalence relation on an n-element set contains all n^2 possible ordered pairs, while the smallest is the identity relation with exactly n ordered pairs. Thus option (B) n^2 and n is correct.