   Chapter 8.1, Problem 19ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Exercises 19-20 refer to unions and intersections of relations. Since relations are subsets of Cartesian products, their unions and intersections can be calculated as for any subsets. Given two relations R and S from A to B,   R   ∪   s = { ( x , y ) ∈   A × B |   ( x , y )   ∈   R   o r ( x , y )   ∈   S }    R   ∩   s = { ( x , y ) ∈   A × B |   ( x , y )   ∈   R   o r ( x , y )   ∈   S } Let A = { 2 , 4 } and B= {6,8,10} and define relations R and S from A to B as follows: For every   ( x , y )   ∈   A × B ,    x ​   R   y   ⇔   x |   y ​   a n d x   S   y ⇔   y − 4 = x . State explicitly which ordered pairs are in A × B ,   R , S , R ∪ S ,     a n d   R ∩ S .

To determine

To find:

Explicitly the ordered pairs in A × B, R, S,R ∪ S, and R n S.

Explanation

Given information:

Let A = {2, 4} and B = {6, 8, 10} and define relations R and S from A to B as follows: For all (x, y) ∈ A × B ,

xRyx|y and

xSyy4=x

Calculation:

A×B consists of all the ordered pairs (x,y) where x is from set A and y is from set B.

R consists all the ordered pairs (x,y) where x|y (i.e. x divides into y ).

S consists of all the ordered pairs (x,y) where y − 4 = x (i.e. y is 4 more than x ).

Since A={2,4} and B={4,6,8}, so

A×B={(2,6),(2,8),(2,10),(4,6),(4,8),(4,10)}R={(2,6),(2,8),(2,10),(4,8)}S={(2,6),(4,8)}

RS will contain all elements of R and S, common elements will be counted once.

RS will contain the coomon elements of R and S

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