Discrete Math Problems

Section 8.1

#11

Let A = {3,4,5} and B= {4,5,6} and let S be the "divides" relation. That is, for all (x,y) IN A * B, x S y <-> x|y

State explicitly which ordered pairs are in S and S⁻¹

#17

Let A={2,3,4,5,6,7,8} and define a relation T on A as follows: For all xy, y IN A, x T y <-> 3|(x-y).

Section 8.2

#14

determine whether the given relation is reflexive, symmetric, transitive or none. Justify.

O is the relation defined on Z as follows: For all m, n IN Z, m O n <-> m-n is odd.

#19

Define a relation I on R as follows: For all real numbers x and y, x I y <-> x-y is irrational.

#29

Let A = R * R. A relation S is defined on A as follows:

For all (x₁,y₁) and (x₂,y₂) in A,

(x₁,y₁) S (x₂,y₂) <-> y₁ = y₂

Section 8.3

#10

the relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R.

A = {-5,-4,-3,2,-1,0,1,2,3,4,5}. R is defined on A as follows: For all m,n IN Z,

m R n <-> 3|(m²-n²)

#23

prove that the relation is an equivalence relation, describe the distinct equivalence classes of each relation.

R is the relation defined on Z as follows: For all (m,n) IN Z, m R n <-> 4|(m²-n²).

#29

Let A be the set of points in the rectangle with x and y coordinates between 0 and 1. That is,

A={(x,y) IN R * R|0<=x<=1 and 0 <=y<=1}.

Define a relation R on A as follows: For all (x₁,y₁) and (x₂,y₂) in A

(x₁,y₁) R (x₂,y₂) <->

(x₁,y₁)=(x₂,y₂) or..

x₁ = 0 and x₂= 1 and y₁ = y₂ or
x₁=1 and x₂=0 and y₁=y₂; or
y₁=0 and y₂=1 and x₁=x₂; or
y₁=1 and y₂=0 and x₁=x₂

In other words, points along the top edge of the rectangle are related to the points along the bottom edge directly beneath them, and all points directly opposite each other along the left and right edges are related to each other. The points in the interior are not related to anything other than themselves. Then R is an equivalence relation on S. Imagine gluing together all the points that are in the same equivalence class. Describe the resulting figure.

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.