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Q1E

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Discrete Mathematics and its Applications

Found in: Page 589

Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

List the triples in the relation\(\{ (a,b,c)|a,b\;{\bf{and}}\;\;c\,{\bf{are}}{\rm{ }}{\bf{integers}}{\rm{ }}{\bf{with}}\;0 < a < b < c < 5\} \).

The resultant answer is \(\{ (1,2,3),(1,2,4),(1,3,4),(2,3,4)\} \).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

\(R = \{ (a,b,c)\mid a,b,c\) are integers with \(0 < a < b < c < 5\} \).

Step 2: Concept ofsets

The concept of set is a very basic one. It is simple; yet, it suffices as the basis on which all abstract notions in mathematics can be built. \(A\) set is determined by its elements. If \(A\) is a set, write \(x \in A\) to say that \(x\) is an element of \(A\).

Step 3: Simplify the expression

This \(0 < a < b < c < 5\) implies that \(a\) can only take on the value 1 or 2 as \(b\) and \(c\) need to be larger than \(a\) while also smaller than 5. When \(a = 1\), then \(b\) can be 2 or 3 as \(b\) needs to be larger than \(a\) and \(c\) need to be larger than \(b\) while also smaller than 5.

When \(a = 1\) and \(b = 2\), then \(c\) can take on the value 3 or 4 (as \(c\) needs to be smaller 5 and larger than \(b\) )

\(\begin{array}{l}(1,2,3) \in R\\(1,2,4) \in R\end{array}\)

When \(a = 1\) and \(b = 3\), then \(c\) can only take on the value of 4 (as \(c\) needs to be smaller 5 and larger than \(b\) ) \((1,3,4) \in R\)

When \(a = 2\), then \(b = 3\) and \(c = 4\) (as \(c\) needs to be smaller 5 and larger than \(b\), while \(b\) is larger than \(a\) ) \((2,3,4) \in R\)

The relation \(R\) is then the set of all found triples: \(R = \{ (1,2,3),(1,2,4),(1,3,4),(2,3,4)\} \).

Most popular questions for Math Textbooks

Let \(S\) be a set with \(n\) elements and let \(a\) and \(b\) be distinct elements of \(S\). How many relations \(R\) are there on \(S\) such that

a) \((a,b) \in R\) ?

b) \((a,b) \notin R\) ?

c) no ordered pair in \(R\) has \(a\) as its first element?

d) at least one ordered pair in \(R\) has \(a\) as its first element?

e) no ordered pair in \(R\) has \(a\) as its first element or \(b\) as its second element?

f) at least one ordered pair in \(R\) either has \(a\) as its first element or has \(b\) as its second element?

Which relations in Exercise $6$ are irreflexive?

(a) To find Relation \({R_1} \cup {R_2}\).

(b) To find Relation \({R_1} \cap {R_2}\).

(c) To find Relation \({R_1} - {R_2}\).

(d) To find Relation \({R_2} - {R_1}\).

(e) To find Relation \({R_1} \oplus {R_2}\).

Adapt Algorithm 1 to find the reflexive closure of the transitive closure of a relation on a set with \(n\) elements.

To prove that the relation \(R\) on set \(A\) is reflexive, if and only if the complementary relation is irreflexive.

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