Discrete Mathematics And Its Applications 7th Edition

7th Edition

ISBN: 9781259152153

Author: Kenneth H. Rosen

Publisher: MCG CUSTOM

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Chapter 9.3, Problem 12E

How can the matrix for R − 1 , the inverse of the relation R, be found from the matrix representing R, when R is a relation on a finite setA?

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Chapter 9.3, Problem 11E

Chapter 9.3, Problem 13E

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By considering appropriate series expansions, e². e²²/2. e²³/3. .... = = 1 + x + x² + · ... when |x| < 1. By expanding each individual exponential term on the left-hand side the coefficient of x- 19 has the form and multiplying out, 1/19!1/19+r/s, where 19 does not divide s. Deduce that 18! 1 (mod 19).

Proof: LN⎯⎯⎯⎯⎯LN¯ divides quadrilateral KLMN into two triangles. The sum of the angle measures in each triangle is ˚, so the sum of the angle measures for both triangles is ˚. So, m∠K+m∠L+m∠M+m∠N=m∠K+m∠L+m∠M+m∠N=˚. Because ∠K≅∠M∠K≅∠M and ∠N≅∠L, m∠K=m∠M∠N≅∠L, m∠K=m∠M and m∠N=m∠Lm∠N=m∠L by the definition of congruence. By the Substitution Property of Equality, m∠K+m∠L+m∠K+m∠L=m∠K+m∠L+m∠K+m∠L=°,°, so (m∠K)+ m∠K+ (m∠L)= m∠L= ˚. Dividing each side by gives m∠K+m∠L=m∠K+m∠L= °.°. The consecutive angles are supplementary, so KN⎯⎯⎯⎯⎯⎯∥LM⎯⎯⎯⎯⎯⎯KN¯∥LM¯ by the Converse of the Consecutive Interior Angles Theorem. Likewise, (m∠K)+m∠K+ (m∠N)=m∠N= ˚, or m∠K+m∠N=m∠K+m∠N= ˚. So these consecutive angles are supplementary and KL⎯⎯⎯⎯⎯∥NM⎯⎯⎯⎯⎯⎯KL¯∥NM¯ by the Converse of the Consecutive Interior Angles Theorem. Opposite sides are parallel, so quadrilateral KLMN is a parallelogram.

By considering appropriate series expansions, ex · ex²/2 . ¸²³/³ . . .. = = 1 + x + x² +…… when |x| < 1. By expanding each individual exponential term on the left-hand side and multiplying out, show that the coefficient of x 19 has the form 1/19!+1/19+r/s, where 19 does not divide s.

Chapter 9 Solutions

Discrete Mathematics And Its Applications 7th Edition

Ch. 9.1 - t the ordered pairs in the...Ch. 9.1 - a) List all the ordered pairs in the relation R =...Ch. 9.1 - each of these relations on the set {1, 2, 3, 4},...Ch. 9.1 - ermine whether the relationRon the set of all...Ch. 9.1 - ermine whether the relationRon the set of all Web...Ch. 9.1 - ermine whether the relationRon the set of all real...Ch. 9.1 - ermine whether the relationRon the set of all...Ch. 9.1 - w that the relationR=Oon a nonempty set S is...Ch. 9.1 - Show that the relationR=on the empty setS=is...Ch. 9.1 - e an example of a relation on a set that is a)...

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How can the matrix for R − 1 , the inverse of the relation R, be found from the matrix representing R, when R is a relation on a finite set A ?

Solution Summary: The author explains that the matrix R-1 represents the inverse of the relation R whose matrix is

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Chapter 9 Solutions