/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Using the boolean matrices $$ ... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 8

Using the boolean matrices $$ A=\left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right], B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right], \text { and } C=\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right] $$ find each. $$(A \odot B) \odot C$$

Short Answer

Expert verified
The short answer is: \((A \odot B) \odot C = \left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \\ \end{array}\right]\).

Step by step solution

01

Compute \(A \odot B\)

To compute the boolean matrix product of A and B, we perform the boolean matrix multiplication operation element-wise: $$ \begin{aligned} (A \odot B) &=\left[\begin{array}{ll} (1 \wedge 0) \vee (1 \wedge 1) & (1 \wedge 1) \vee (1 \wedge 0)\\ (0 \wedge 0) \vee (0 \wedge 1) & (0 \wedge 1) \vee (0 \wedge 0)\\ \end{array}\right] \\ &= \left[\begin{array}{ll} (0) \vee (1) & (1) \vee (0) \\ (0) \vee (0) & (0) \vee (0) \\ \end{array}\right] \\ &= \left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \\ \end{array}\right] \end{aligned} $$ So, \(A \odot B = \left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \\ \end{array}\right]\).
02

Compute \((A \odot B) \odot C\)

Now let's compute the boolean matrix product of \(A \odot B\) and C: $$ \begin{aligned} (A \odot B) \odot C &= \left[\begin{array}{ll} (1 \wedge 0) \vee (1 \wedge 1) & (1 \wedge 0) \vee (1 \wedge 0) \\ (0 \wedge 0) \vee (0 \wedge 1) & (0 \wedge 0) \vee (0 \wedge 0) \\ \end{array}\right] \\ &= \left[\begin{array}{ll} (0) \vee (1) & (0) \vee (0) \\ (0) \vee (0) & (0) \vee (0) \\ \end{array}\right] \\ &= \left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \\ \end{array}\right] \end{aligned} $$ So, (A 鈯� B) 鈯� C = \(\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \\ \end{array}\right]\).

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Most popular questions from this chapter

When is a relation on a set \(A\) not: Reflexive?

Find the partition of the set \(\\{a, b, c\\}\) induced by each equivalence relation. $$[(a, a),(b, b),(c, c)\\}$$

The number of partitions of a set with size \(n\) is also given by the Bell number \(B_{n} .\) Using Bell numbers, compute the number of partitions of a set with each of the sizes in Exercises \(33-36.\)

In Exercises \(17-19,\) the adjacency matrices of three relations on \(\\{a, b, c\\}\) are given. Determine if each relation is reflexive, symmetric, or antisymmetric. $$ \left[\begin{array}{lll}{0} & {1} & {1} \\ {1} & {0} & {1} \\ {1} & {0} & {0}\end{array}\right] $$

Let \(R\) be any relation on a set \(A .\) Prove each. RU \(\Delta\) is the smallest reflexive relation containing \(R\) (Hint: Assume there is a reflexive relation \(S\) such that \(R \subseteq S \subseteq R \cup \Delta\) . Prove that \(S=R\) or \(S=R \cup \Delta\) .)
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