Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

Show that if \(A\), \(B\) and \(C\) are sets, then \(\overline {A \cap B \cap C} = \overline A \cup \overline B \cup \overline C \).

(a) by showing each side is a subset of the other side.

(b) using a membership table.

Suppose that A, B and C are sets such that \(A \subseteq B\) and \(B \subseteq C\). Show that \(A \subseteq C\).

Let A and B be two $\mathit{n}\mathbf{\times }\mathit{n}$matrices. Show that

a) $\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}{\mathbf{)}}^{\mathbf{t}}\mathbf{=}{\mathit{A}}^{\mathbf{t}}\mathbf{+}{\mathit{B}}^{\mathbf{t}}$.

b) $\mathbf{(}\mathit{A}\mathit{B}{\mathbf{)}}^{\mathbf{t}}\mathbf{=}{\mathit{B}}^{\mathbf{t}}{\mathit{A}}^{\mathbf{t}}$.

If A and B are $\mathit{n}\mathbf{\times }\mathit{n}$ matrices with $\mathit{A}\mathit{B}\mathbf{=}\mathit{B}\mathit{A}\mathbf{=}{\mathit{I}}_{\mathbf{n}}$, then B is called the inverse of A (this terminology is appropriate because such a matrix B is unique) and A is said to be invertible. The notation $\mathit{B}\mathbf{=}{\mathit{A}}^{\mathbf{-}\mathbf{1}}$ denotes that B is the inverse of A.

Consider these functions from the set of students in a discrete mathematics class. Under what conditions is the function one-to-one if it assigns to a teacher his or her

1. Office
2. Assigned bus to chaperone in a group of buses taking students on a field trip
3. Salary
4. Social security number

Define the product of two matrices A and B. When is this product defined?

Prove that if f and g are functions from A to B and $S_f=S_g$, the $f\left(x\right)=g\left(x\right)\forall x\in A$.

Show that$\left[\begin{array}{cc}2& 3\\ 1& 2\\ -1& -1\end{array}\begin{array}{c}-1\\ 1\\ 3\end{array}\right]$is the inverse ofrole="math" localid="1668443022252" $\left[\begin{array}{ccc}7& 8& 5\\ 4& 5& -3\\ 1& -1& 1\end{array}\right]$ .

Find the two sets A and B such that\(A \in B\)and\(A \subseteq B\).

Let A, B, and C is set.Show that

a) \(\left( {A \cup B} \right) \subseteq \left( {A \cup B \cup C} \right)\)

b) \(\left( {A \cap B \cap C} \right) \subseteq \left( {A \cap B} \right)\)

c) \(\left( {A - B} \right) - C \subseteq A - C\)

d)\(\left( {A - C} \right) \cap \left( {C - B} \right) = \phi \)

e) \(\left( {B - A} \right) \cup \left( {C - A} \right) = \left( {B \cup C} \right) - A\)

Show that if A andBare sets|A| = |B|then P (A) = P (B) .