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Readability settings Prev Up Next \(\require{cancel}
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Activities 9.8 Activities
Activity 9.1 .
For each member of your group, consider the set of all math and computing science courses you have taken so far at university. What is the intersection of these sets for your group?
Activity 9.2 .
Is it possible to have two sets
\(A \) and
\(B \) with
\(A \union B = A \intersection B \text{?}\)
Activity 9.3 . Cancellation is not always valid.
(a) Demonstrate using an example that
\((A \union B) \relcmplmnt B = A \) is
not a valid simplification in set theory.
(b) Demonstrate using an example that
\(A \union B = A \union C \lgcimplies A = B \) is
not a valid simplification in set theory.
Activity 9.4 .
Fill in the blank with a concept from the reading.
Breaking the students in a class into groups is an example of
.
Activity 9.5 .
(a) (b)
Activity 9.6 .
Recall the following notation and terminology from linear algebra.
\(\matrixring_n(\R) \) represents the set of all \(n \times n \) matrices.
Scalar matrix means a scalar multiple of the identity matrix.
Singular matrix means not invertible.
Let
\(V \) and
\(S \) represent subsets of
\(\matrixring_n(\R) \text{,}\) where
\(V \) consists of the
invertible \(n \times n \) matrices, and
\(S \) consists of the
scalar \(n \times n \) matrices. Also, as usual write
\(\zerovec \) for the
\(n \times n \) zero matrix.
Express each of the following statements using the symbols of set theory:
\begin{equation*}
\in, \;\;\;
\subseteq, \;\;\;
\union, \;\;\;
\intersection, \;\;\;
\emptyset, \;\;\;
\text{etc.}
\end{equation*}
(a)
\(\zerovec \) is a scalar matrix.
(b)
\(\zerovec \) is scalar and singular.
(c)
\(\zerovec \) is the
only scalar, singular matrix.
(d) Every scalar matrix besides
\(\zerovec \) is invertible.
(e) Every matrix is either invertible or singular.
Activity 9.7 .
Pick another group in the class and list the elements of the Cartesian product of your group with that other group. If that group happened to also choose your group for this task, would their answer be the same as yours?
Activity 9.8 .
List the elements of the power set of your group. Make sure you have all the
\(\{\ \} \) -pairs you need in all the right places.
Activity 9.9 .
What is
\(A \cartprod \emptyset \text{?}\) What about
\(\emptyset \cartprod B \text{?}\)
Activity 9.10 .
For alphabet
\(\Sigma = \{\mathrm{a},\mathrm{b},\mathrm{c}\} \text{,}\) describe the elements of
\(\words{\Sigma} \) and
\(\words{(\words{\Sigma})} \text{:}\)
Elements of
\(\words{\Sigma} \) are
.
Elements of
\(\words{(\words{\Sigma})} \) are
.
Is the equality of sets
\(\words{(\words{\Sigma})} = \words{\Sigma} \) true?
Activity 9.11 .
The equality of sets
\begin{equation*}
A \cartprod (B \relcmplmnt C) = (A \cartprod B) \relcmplmnt (A \cartprod C)
\end{equation*}
is true in general.
Activity 9.12 .
The equality of sets
\((A \cartprod B) \union (C \cartprod D) = (A \union C) \cartprod (B \union D) \) is false in general.
(a) Write down definitions for example sets
\(A,B,C,D \) that form a counterexample.
(b) Can you come up with some conditions on
\(A,B,C,D \) that make this equality true?
Activity 9.13 .
Write a formal proof of the equality
\begin{equation*}
\powset{A \intersection B} = \powset{A} \intersection \powset{B}
\end{equation*}
Activity 9.14 .
Informally explain why the set equality
\(\powset{A \union B} = \powset{A} \union \powset{B} \) is not true in general.
