Chapter 2: Q28E (page 185)

Let $A = [\begin{matrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$
Find
a) $A^{[2]}$ .
b) $A^{[3]}$
c) $A \lor A^{[2]} \lor A^{[3]}$ .

Short Answer

Expert verified

a) Required answer is $A^{[2]} = [\begin{array}{lll} 1 0 0 \\ 1 1 0 \\ 1 0 1 \end{array}]$ .

b) Required answer is $A^{[3]} = [\begin{array}{lll} 1 0 0 \\ 1 0 1 \\ 1 1 0 \end{array}]$

c) Required answer is $A \lor A^{[2]} \lor A^{[3]} = [\begin{array}{lll} 1 0 0 \\ 1 1 1 \\ 1 1 1 \end{array}]$ .

Step by step solution

Step 1:

(a) $A^{[2]}$ is the boolean product of A with itself, thus $A ⊙ A$ .

The ijth element of the Boolean product $A ⊙ B$ takes the disjunction of all conjunctions of corresponding elements A and B in the row of $i^{t h}$ and in the $j^{t h}$ column of B.

$A^{[2]} = A ⊙ A$

$= [\begin{array}{lll} (1 \land 1) \lor (0 \land 1) \lor (0 \land 0) (1 \land 0) \lor (0 \land 0) \lor (0 \land 1) (1 \land 0) \lor (0 \land 1) \lor (0 \land 0) \\ (1 \land 1) \lor (0 \land 1) \lor (1 \land 0) (1 \land 0) \lor (0 \land 0) \lor (1 \land 1) (1 \land 0) \lor (0 \land 1) \lor (1 \land 0) \\ (0 \land 1) \lor (1 \land 1) \lor (0 \land 0) (0 \land 0) \lor (1 \land 0) \lor (0 \land 1) (0 \land 0) \lor (1 \land 1) \lor (0 \land 0) \end{array}]$

$b_{1} Λ b_{2}$ is equal to 1 if $b_{1} = b_{2} = 1$

$= [\begin{array}{lll} 1 \lor 0 \lor 0 0 \lor 0 \lor 0 0 \lor 0 \lor 0 \\ 1 \lor 0 \lor 0 0 \lor 0 \lor 1 0 \lor 0 \lor 0 \\ 0 \lor 1 \lor 0 0 \lor 0 \lor 0 0 \lor 1 \lor 0 \end{array}]$

$b_{1} \lor b_{2} \lor b_{3}$ is equal to 1 if $b_{1} = 1$ or $b_{2} = 1$ or $b_{3} = 1$

$= [\begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{matrix}]$

Step 2:

(b) $A^{[3]}$ is the Boolean product of $A^{[2]}$ with A, thus $A^{[2]} ⊙ A$ .

The ijth element of the Boolean product $A ⊙ B$ takes the disjunction of all conjunctions of corresponding elements Aand B in the ith row of and in the jth column of B.

$A^{[3]} = A^{[2]} ⊙ A$

$= [\begin{array}{lll} (1 \land 1) \lor (0 \land 1) \lor (0 \land 0) (1 \land 0) \lor (0 \land 0) \lor (0 \land 1) (1 \land 0) \lor (0 \land 1) \lor (0 \land 0) \\ (1 \land 1) \lor (1 \land 1) \lor (0 \land 0) (1 \land 0) \lor (1 \land 0) \lor (0 \land 1) (1 \land 0) \lor (1 \land 1) \lor (0 \land 0) \\ (1 \land 1) \lor (0 \land 1) \lor (1 \land 0) (1 \land 0) \lor (0 \land 0) \lor (1 \land 1) (1 \land 0) \lor (0 \land 1) \lor (1 \land 0) \end{array}]$

$b_{1} \land b_{2}$ is equal to 1 if $b_{1} = b_{2} = 1$

$= [\begin{array}{lll} 1 \lor 0 \lor 0 0 \lor 0 \lor 0 0 \lor 0 \lor 0 \\ 1 \lor 1 \lor 0 0 \lor 0 \lor 0 0 \lor 1 \lor 0 \\ 1 \lor 0 \lor 0 0 \lor 0 \lor 1 0 \lor 0 \lor 0 \end{array}]$

$b_{1} \lor b_{2} \lor b_{3}$ is equal to 1 if $b_{1} = 1$ or $b_{2} = 1$ or $b_{3} = 1$

data-custom-editor="chemistry" $= [\begin{matrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix}]$

Step 3:

$\begin{aligned} A \lor A^{[2]} \lor A^{[3]} = [\begin{array}{lll} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}] \lor [\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array}] \lor [\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}] \\ = [\begin{array}{lll} 1 \lor 1 \lor 1 & 0 \lor 0 \lor 0 & 0 \lor 0 \lor 0 \\ 1 \lor 1 \lor 1 & 0 \lor 1 \lor 0 & 1 \lor 0 \lor 1 \\ 0 \lor 1 \lor 1 & 1 \lor 0 \lor 1 & 0 \lor 1 \lor 0 \end{array}] \end{aligned}$

$b_{1} \lor b_{2}$ is equal to 1 ifrole="math" localid="1668491140956" $b_{1} = 1$ or $b_{2} = 1$

$= [\begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$

Let $A = [\begin{matrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$
Find
a) $A^{[2]}$ .
b) $A^{[3]}$
c) $A \lor A^{[2]} \lor A^{[3]}$ .

Short Answer

Step by step solution

Step 1:

Step 2:

Step 3:

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LetA=100101010Finda) A[2].b)A[3]c) A∨A[2]∨A[3].

Short Answer

Step by step solution

Step 1:

Step 2:

Step 3:

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Probability and Statistics

Statistics

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Discrete Mathematics

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Let $A = [\begin{matrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}]$
Find
a) $A^{[2]}$ .
b) $A^{[3]}$
c) $A \lor A^{[2]} \lor A^{[3]}$ .