Exercises 11.3 Exercises
Given these matrices, answer the following questions or compute the following matrices (if they exist).
1.
Calculate \(A \vee C\text{.}\)
2.
Calculate \(C \wedge A\text{.}\)
3.
Calculate \(A \wedge B\text{.}\)
4.
Calculate \(A \odot B\text{.}\)
5.
Calculate \(B \odot A\text{.}\)
6.
Calculate \(D \odot B\text{.}\)
7.
Calculate \(D^{[2]}\) (which is the same as \(D \odot D\)).
8.
Calculate
or say the matrix doesn't exist.
9.
Calculate
or say the matrix doesn't exist.
10.
Calculate
or say the matrix doesn't exist.
11.
Calculate
12.
Let \(\Sigma = \{0,1\}\text{,}\) \(n\) be a positive integer, and let \(\Sigma^n\) be the set of all bit strings of length \(n\text{.}\) Describe a Boolean algebra structure on \(\Sigma^n\text{.}\) In other words, define the meet, join, and complement operations and give examples, and tell what the bottom and top objects are.
The complement of a string is the string with each \(1\) replaced by a \(0\) and vice-versa, e.g. \(\neg 0101 = 1010\text{.}\)
The meet of two strings has a \(1\) only in the digits where both strings had a \(1\) and \(0\)s elsewhere, e.g. \(1101 \wedge 0110 = 0100\text{.}\)
The join of two strings has a \(1\) anywhere at least one of the two strings has a \(1\text{,}\) e.g. \(10110 \vee 00101 = 10111\text{.}\)
The top element is the string of all ones and the bottom element is the string of all zeros.