Explain the relationship between equivalence relations on a set and partitions of this set.

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)?

Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs
(0, 1),(1, 1),(1, 2),(2, 0),(2, 2),(3, 0). Find reflexive, symmetric and transitive closure of R.

Can someone explain equivalence classes in discrete math?

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 R be the relation {(1, 2), (1, 3), (2, 3), (2, 4), (3, 1)}, and let S be the relation {(2, 1), (3, 1), (3, 2), (4, 2)}.

Find S ◦ R.

Consider the equivalence relation R={(x,y)|c-y is an integer}. a) What is the equivalence class of 1 for this equivalence relation? b) What is the equivalence class of 1/2 for this equivalence relation?

a) Define (using truth tables) the disjunction, conjunction, exclusive or, conditional, and biconditional of the propositions p and q. b) What are the disjunction, conjunction, exclusive or, conditional, and biconditional of the propositions “I’ll go to the movies tonight” and “I’ll finish my discrete mathematics homework”?

Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if a) x ≠ y. b) xy ≥ 1. c) x = y + 1 or x = y − 1. d) x ≡ y (mod 7). e) x is a multiple of y. f ) x and y are both negative or both nonnegative. g) x = y². h) x ≥ y².

a) Show that congruence modulo m is an equivalence relation whenever m is a positive integer. b) Show that the relation {(a, b) | a ≡ ±b (mod 7)} is an equivalence relation on the set of integers.

Prove that a union of any two countably infinite sets is countably infinite.

Let the universal set be {A, B, C, D, F} and let G{(A, 1), (B, 0.7), (C, 0.4), (D, 0.1), (F, 0)} be a fuzzy set defined by Greg to describe what he feels is a good grade. Determine G'.

Let U={a, b, c, d, e, f, g}, A={a, b, f, g}, B={c, d, e}, C={a, g}, and D={a, b, c, d, e, f}. Use the roster method to write each of the following sets. A'

Give an example of two uncountable sets A and B such that A − B is a) finite. b) countably infinite. c) uncountable.