Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Let \({R_1} = \{ (1,2),(2,3),(3,4)\} \) and \({R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),\)\((3,1),(3,2),(3,3),(3,4)\} \) be relations from \(\{ 1,2,3\} \) to \(\{ 1,2,3,4\} \). Find
a) \({R_1} \cup {R_2}\).
b) \({R_1} \cap {R_2}\).
c) \({R_1} - {R_2}\).
d) \({R_2} - {R_1}\).

(a) \({R_1} \cup {R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4)\} \)

(b) \({R_1} \cap {R_2} = \{ (1,2),(2,3),(3,4)\} \)

(d) \({R_2} - {R_1} = \{ (1,1),(2,1),(2,2),(3,1),(3,2),(3,3)\} \)

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given Data

\(\begin{array}{l}{R_1} = \{ (1,2),(2,3),(3,4)\} \\{R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4)\} \end{array}\)

Step 2: Concept of the union, intersection and difference

Union \(A \cup B\): All elements that are either in \(A\)or in \(B\)

Intersection \(A \cap B\): All elements that are both in \(A\)and in \(B\).

Difference \(A - B\): All elements in \(A\) that are NOT in \(B\)

Step 3: Determine the value of \({R_1} \cup {R_2}\)

(a)

The union of two relations contains all ordered pairs that are in either relation.

\({R_1} \cup {R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4)\} \)

Step 4: Determine the value of \({R_1} \cap {R_2}\)

(b)

The intersection of two relations contains all ordered pairs that are in both relations. We note that all ordered pairs in \({R_1}\)also occur in \({R_2}\).

\({R_1} \cap {R_2} = \{ (1,2),(2,3),(3,4)\} \)

Step 5: Determine the value of \({R_1} - {R_2}\)

(c)

\({R_1} - {R_2}\)contains all ordered pairs that are in the relation \({R_1}\) that do not occur in the relation \({R_2}\). We note that all ordered pairs in \({R_1}\) also occur in \({R_2}\), thus the difference \({R_1} - {R_2}\) does not contain any elements.

\({R_1} - {R_2} = \emptyset \)

Step 6: Determine the value of \({R_2} - {R_1}\)

(d)

\({R_2} - {R_1}\) contains all ordered pairs that are in the relation \({R_1}\)that do not occur in the relation \({R_2}\).

\({R_2} - {R_1} = \{ (1,1),(2,1),(2,2),(3,1),(3,2),(3,3)\} \)

Therefore,

(a) \({R_1} \cup {R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4)\} \)

(b) \({R_1} \cap {R_2} = \{ (1,2),(2,3),(3,4)\} \)

(d) \({R_2} - {R_1} = \{ (1,1),(2,1),(2,2),(3,1),(3,2),(3,3)\} \)

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Discrete Mathematics and its Applications

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Short Answer

Let \({R_1} = \{ (1,2),(2,3),(3,4)\} \) and \({R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),\)\((3,1),(3,2),(3,3),(3,4)\} \) be relations from \(\{ 1,2,3\} \) to \(\{ 1,2,3,4\} \). Find
a) \({R_1} \cup {R_2}\).
b) \({R_1} \cap {R_2}\).
c) \({R_1} - {R_2}\).
d) \({R_2} - {R_1}\).

Step by Step Solution

Step 1: Given Data

Step 2: Concept of the union, intersection and difference

Step 3: Determine the value of \({R_1} \cup {R_2}\)

Step 4: Determine the value of \({R_1} \cap {R_2}\)

Step 5: Determine the value of \({R_1} - {R_2}\)

Step 6: Determine the value of \({R_2} - {R_1}\)

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Discrete Mathematics and its Applications

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Short Answer

Let \({R_1} = \{ (1,2),(2,3),(3,4)\} \) and \({R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),\)\((3,1),(3,2),(3,3),(3,4)\} \) be relations from \(\{ 1,2,3\} \) to \(\{ 1,2,3,4\} \). Finda) \({R_1} \cup {R_2}\).b) \({R_1} \cap {R_2}\).c) \({R_1} - {R_2}\).d) \({R_2} - {R_1}\).

Step by Step Solution

Step 1: Given Data

Step 2: Concept of the union, intersection and difference

Step 3: Determine the value of \({R_1} \cup {R_2}\)

Step 4: Determine the value of \({R_1} \cap {R_2}\)

Step 5: Determine the value of \({R_1} - {R_2}\)

Step 6: Determine the value of \({R_2} - {R_1}\)

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Pure Maths

Let \({R_1} = \{ (1,2),(2,3),(3,4)\} \) and \({R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),\)\((3,1),(3,2),(3,3),(3,4)\} \) be relations from \(\{ 1,2,3\} \) to \(\{ 1,2,3,4\} \). Find
a) \({R_1} \cup {R_2}\).
b) \({R_1} \cap {R_2}\).
c) \({R_1} - {R_2}\).
d) \({R_2} - {R_1}\).