Define three equivalence relations on the set of students in your discrete mathematics class different from the relations discussed in the text. Determine the equivalence classes for each of these equivalence relations.

Let A be the set of students at your school and B the set of books in the school library. Let R₁ and R₂ be the relations consisting of all ordered pairs (a, b), where student a is required to read book b in a course, and where student a has read book b, respectively. Describe the ordered pairs in each of these relations. a) R₁ ∪ R₂ b) R₁ ∩ R₂ c) R₁ ⊕ R₂ d) R₁ − R₂ e) R₂ − R₁

Show that the relation of logical equivalence on the set of all compound propositions is an equivalence relation. What are the equivalence classes of F and of T?

Let R be the relation on the set of all sets of real numbers such that S R T if and only if S and T have the same cardinality. Show that R is an equivalence relation. What are the equivalence classes of the sets {0, 1, 2} and Z?

Which of these collections of subsets are partitions of {1, 2, 3, 4, 5, 6}? a) {1, 2},{2, 3, 4},{4, 5, 6} b) {1}, {2, 3, 6}, {4}, {5} c) {2, 4, 6}, {1, 3, 5} d) {1, 4, 5}, {2, 6}

Show that the relation R consisting of all pairs (x, y) such that x and y are bit strings of length three or more that agree except perhaps in their first three bits is an equivalence relation on the set of all bit strings of length three or more.

Give an argument using rules of inference to show that the conclusion follows from the hypotheses. Hypotheses: Everyone in the discrete mathematics class loves proofs. Someone in the discrete mathematics class has never taken calculus. Conclusion: Someone who loves proofs has never taken calculus.

What is the definition of a graph in discrete mathematics?

Which of the following is a negation for “All discrete mathematics students are athletic”? More than one answer may be correct. a. There is a discrete mathematics student who is nonathletic. b. All discrete mathematics students are nonathletic. c. There is an athletic person who is a discrete mathematics student. d. No discrete mathematics students are athletic. e. Some discrete mathematics students are nonathletic. f. No athletic people are discrete mathematics students.

a) Define an equivalence relation. b) Which relations on the set {a, b, c, d} are equivalence relations and contain (a, b) and (b, d)?

State whether the variable is discrete or continuous.

17. The number of mathematics majors in your school

Which of these collections of subsets are partitions of the set of bit strings of length 8? a) the set of bit strings that begin with 1, the set of bit strings that begin with 00, and the set of bit strings that begin with 01 b) the set of bit strings that contain the string 00, the set of bit strings that contain the string 01, the set of bit strings that contain the string 10, and the set of bit strings that contain the string 11 c) the set of bit strings that end with 00, the set of bit strings that end with 01, the set of bit strings that end with 10, and the set of bit strings that end with 11 d) the set of bit strings that end with 111, the set of bit strings that end with 011, and the set of bit strings that end with 00 e) the set of bit strings that contain 3k ones for some nonnegative integer k; the set of bit strings that contain 3k + 1 ones for some nonnegative integer k; and the set of bit strings that contain 3k + 2 ones for some nonnegative integer k.

Which of these relations on the set of all functions from Z to Z are equivalence relations? Determine the properties of an equivalence relation that the others lack. a) {(f, g) | f(1) = g(1)} b) {(f, g) | f(0) = g(0) or f(1) = g(1)} c) {(f, g) | f(x) − g(x) = 1 for all x ∈ Z} d) {(f, g) | for some C ∈ Z, for all x ∈ Z, f(x) − g(x) = C} e) {(f, g) | f(0) = g(1) and f(1) = g(0)}

Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack. a) {(0, 0), (1, 1), (2, 2), (3, 3)} b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)} c) {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} d) {(0, 0), (1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} e) {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2), (3, 3)}