
5. On the set of integers Z define the following relation: "aRb if and only if...
Question 11 Let's define an equivalence relation R on the set of integers by aRb if and only if 5|3a + 7b What is the cardinality of the partition induced by R? Not yet answered Points out of 1.00 P Flag question Select one: a. 1 O b.4 O C. 5 d. 2 O e. 7 O f. infinite
1. Define a relation on Z by aRb provided a -b a. Prove that this relation is an equivalence relation. b. Describe the equivalence classes. 2. Define a relation on Z by akb provided ab is even. Use counterexamples to show that the reflexive and transitive properties are not satisfied 3. Explain why the relation R on the set S-23,4 defined by R - 11.1),(22),3,3),4.4),2,3),(32),(2.4),(4,2)) is not an equivalence relation.
8.) Consider the integers Z. Dene the relation on Z by x y if
and only
if 7j(y + 6x). Prove:
a.) The relation is an equivalence relation.
b.) Find the equivalence class of 0 and prove that it is a subgroup
of Z
with the usual addition operator on the integers.
8.) Consider the integers Z. Define the relation ~ on Z by x ~ y if and only if 7)(y + 6x). Prove: a.) The relation is an...
Discrete Math. Show all steps clearly
Define a relation R on the set of all integers Z as follows: Is R a partial order relation? Prove or give a counterexample.
[12] 5. Let A = {1, 2, 3, 4, ..., 271}. Define the relation R on A x A by: for any (a,b), (c,d) E AXA, (a,b) R (c,d) if and only if a +b=c+d. (a) Prove that R is an equivalence relation on AX A. (b) List all the elements of [(3,3)], the equivalence class of (3, 3). (c) How many equivalence classes does R have? Explain. (d) Is there an equivalence class that has exactly 271 elements? Explain.
16. (8 points) Let Z be the integers and let A - Zx Z. Define the relation R on A by (a, b) R(c, d) if and only if a c and b 3 d for all (a, b), (c, d)E A. Prove that R is a partial ordering on A that is not a total ordering.
16. (8 points) Let Z be the integers and let A - Zx Z. Define the relation R on A by (a, b)...
Define the set F- (XI X is a finite set of counting numbers) and the relation is a finiice sei of counting nuobors and the relation {(X Z〉 | Ye F and Z € Fand y-2). This relation is just a version of the usual subset relation, but restricted to only apply to the sets in F Prove: CFis a partial order. Prove: Cis not symmetric and connected. Prove: If R is an equivalence relation, it is also a euclidean...
4) Define a relation TC Nx N such that T = {(a,b) a EA A DEA 18- b = 2c+1 for some integer c}. (N is the set of non-negative integers.) a) Prove that this relation is not reflexive. b) Prove that this relation is symmetric. c) Define the term anti-transitive as the following: Given a set A and a relation R, if for all a,b,ceA, (aRb a bRc A cRa) = (a = b v b= c) Prove that...
Let S = Z and R be the relation defined by R = {Z times Z - (n, n)|n Element Z}. (a) Define the relation R, that is aRb if and only if ..... (b) Prove that R^2 = Z times Z
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the...