Exercises 4-3 deal with the Boolean algebra {0, 1} with addition, multiplication, and complement defined at the ginning of the section. In each case, use a table as in Example 8. 23. Verify the zero property. The Boolean operator ⊕ , called the XOR operator, is defined by 1 ⊕ 1 = 0 , 1 ⊕ 0 = 1 , 0 ⊕ 1 = 1 , and 0 ⊕ 0 = 0 .
Exercises 4-3 deal with the Boolean algebra {0, 1} with addition, multiplication, and complement defined at the ginning of the section. In each case, use a table as in Example 8. 23. Verify the zero property. The Boolean operator ⊕ , called the XOR operator, is defined by 1 ⊕ 1 = 0 , 1 ⊕ 0 = 1 , 0 ⊕ 1 = 1 , and 0 ⊕ 0 = 0 .
Solution Summary: The author explains that the zero property is verified using the Boolean algebra left0,1right.
Exercises 4-3 deal with the Boolean algebra {0, 1} with addition, multiplication, and complement defined at the ginning of the section. In each case, use a table as in Example 8.
23. Verify the zero property.
The Boolean operator
⊕
, called the XOR operator, is defined by
1
⊕
1
=
0
,
1
⊕
0
=
1
,
0
⊕
1
=
1
, and
0
⊕
0
=
0
.
The following are equivalent in a Boolean algebra:(1)a+b=b, (2)a*b=a, (3)a'+b=1, (4)a*b'=0i. Prove the equivalence of (3) and (4) . ii. Prove the equivalence of (1) and (2) .
Consider the following subset $S \subset \mathbb{R}^{4}$ $$ S=\left{\left[\begin{array}{r} 1 \ 0 \ -2 \ 5 \end{array}\right],\left[\begin{array}{r} 2 \ 1 \ 0 \ -1 \end{array}\right],\left[\begin{array}{l} 1 \ 1 \ 2 \ 1 \end{array}\right],\left[\begin{array}{r} -3 \ -4 \ -10 \ 8 \end{array}\right]\right} $$ (a) Is $\operatorname{span}(S)=\mathbb{R}^{4}$ ? Explain your reasoning and show work to justify your answer. (b) Determine if $S$ is linearly dependent or independent. Explain why. (c) i. If possible, write $\left[\begin{array}{r}-2 \ -1 \ 0 \ 6\end{array}\right]$ as a linear combination of vectors in $S$. ii. Is this linear combination unique? Why or why not, explain your reasoning. (d) Find two vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^{4},$ if possible, such that $\mathbf{u} \notin \operatorname{span}(S)$ but $\mathbf{v} \in \operatorname{span}(S) . \quad[3+2]$ Justify your answer.
Chapter 12 Solutions
Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
Using and Understanding Mathematics: A Quantitative Reasoning Approach (6th Edition)
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