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The solutions to quiz 4 of cse 260, fall 2011. It covers the properties of relations, specifically reflexivity, symmetry, and transitivity, as well as the reflexive and symmetric closures of a relation. The quiz questions involve justifying whether certain relations are reflexive, symmetric, and transitive, as well as finding the reflexive and symmetric closures of given relations.
Typology: Quizzes
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(1) The relation R is defined on the set of all real numbers as follows: (x, y) ∈ R if and only if xy ≥ 3. Justify your answers.
(a) Is R reflexive?
No, for example (0, 0) 6 ∈ R, since (0)(0) 6 ≥ 3.
(b) Is R symmetric?
Yes. Assume (x, y) ∈ R. Then xy ≥ 3. But then yx ≥ 3, and so (y, x) ∈ R.
(c) Is R transitive?
No. For example: (1, 10) ∈ R and (10, 1) ∈ R, since (1)(10) = 10 ≥ 3; but (1, 1) 6 ∈ R, since (1)(1) 6 ≥ 3.
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2 QUIZ 4 CSE 260, F
(2) Let R be the relation on the set { 1 , 2 , 3 } represented by the matrix:
where the rows and columns correspond to the integers listed in increasing order.
(a) What matrix represents the relation R ◦ R.
(b) Is R transitive? (Justify your answer.)
No. If it were transitive, then every location containing a 1 in MR◦R would also have a 1 in MR. But the answer to the previous question indicates that (2, 3) ∈ R ◦ R, although (2, 3) 6 ∈ R. (2R1 and 1R3, but its not the case that 2 R3.) Alternately, (1, 1) ∈ R ◦ R, although (1, 1) 6 ∈ R. (1R2 and 2R1, but its not the case that 1R1.) Alternately, (3, 1) ∈ R ◦ R, although (3, 1) 6 ∈ R. (3R 2 and 2R1, but its not the case that 3R1.)
(c) What is the reflexive closure of R? R ∪ {(1, 1), (3, 3)}
(d) Is (2, 3) in the symmetric closure of R? Why or why not?
Yes. Because (3, 2) is in R, it is also in the symmetric closure of R. Since the symmetric closure of R is symmetric and it contains (3, 2), we conclude that (2, 3) is also in the symmetric closure of R. Alternately, the matrix for the symmetric closure of R is:
The 1 in the second row and third column indicates that (2, 3) is in the relation.
