Chapter 3

Let \(A=\left[\begin{array}{ccc}1 & 0 & -1 \\ 0 & 2 & 3\end{array}\right], B=\left[\begin{array}{ccc}0 & -2 & 5 \\ 0 & 0 & 1\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-3 & 0 & 0 \\ 0 & 1 & 2\end{array}\right] .\) Find each. $$3 B-2 C$$

Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lceil x\rceil+\lceil y\rceil$$

Let \(\Sigma=\\{0,1\\} .\) Let \(f: \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x)=\|x\| .\) Evaluate \(f(x)\) for each value of \(x\). $$1010100$$

Let \(A=\left[\begin{array}{ccc}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{ccc}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right]\) . Find each. $$3 A+(-2) B$$

Rewrite each sum using the summation notation. $$1+3+5+\cdots+23$$

Determine if the function \(g\) is the inverse of the corresponding function \(f\). $$f(x)=x^{2}, x \geq 0 ; g(x)=\sqrt{x}, x \geq 0$$

Find the range of each function on \(\mathbb{R}\). $$f(x)=\lfloor x\rfloor+\lfloor-x\rfloor$$

Use the pigeonhole principle to prove the following. If five points are chosen inside a unit square, then the distance between at least two of them is no more than \(\sqrt{2} / 2\) .

Let \(\Sigma=\\{0,1\\}\) . Let \(f : \Sigma^{*} \rightarrow\) W defined by \(f(x)=\|x\| .\) Evaluate \(f(x)\) for each value of \(x .\) 0001000

Determine if each function from \(\mathbb{R}\) to \(\mathbf{Z}\) is surjective. $$g(x)=\lfloor x\rfloor$$

Let \(A=\left[\begin{array}{ccc}1 & 0 & -1 \\ 0 & 2 & 3\end{array}\right], B=\left[\begin{array}{ccc}0 & -2 & 5 \\ 0 & 0 & 1\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-3 & 0 & 0 \\ 0 & 1 & 2\end{array}\right] .\) Find each. $$3 A+(-2) B$$

If five points are chosen inside a unit square, then the distance between at least two of them is no more than \(\sqrt{2} / 2\)

Let \(\Sigma=\\{0,1\\} .\) Let \(f: \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x)=\|x\| .\) Evaluate \(f(x)\) for each value of \(x\). $$0001000$$

Rewrite each sum using the summation notation. $$3^{1}+3^{2}+\cdots+3^{10}$$

Let \(A=\left[\begin{array}{lll}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{rrr}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{rrr}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right] .\) Find each. $$A+B$$

Determine if the function \(g\) is the inverse of the corresponding function \(f\). $$f(x)=x^{2}, x \leq 0 ; g(x)=-\sqrt{x}, x \geq 0$$

Use the pigeonhole principle to prove the following. Five points are chosen inside an equilateral triangle of unit side. The distance between at least two of them is no more than \(1/ 2\) .

Let \(\Sigma=\\{0,1\\}\) . Let \(f : \Sigma^{*} \rightarrow\) W defined by \(f(x)=\|x\| .\) Evaluate \(f(x)\) for each value of \(x .\) 00110011

Let \(A\) be an \(m \times n\) matrix, \(B\) a \(p \times q\) matrix, and \(C\) an \(r \times s\) matrix. Under what conditions is each defined? Find the size of each when defined. (Note: \(A^{2}\) means \(A A .\) ) $$A+B$$

Five points are chosen inside an equilateral triangle of unit side. The distance between at least two of them is no more than \(1 / 2 .\)

Find the range of each function on \(\mathbb{R}\). $$f(x)=\lceil x\rceil+\lceil-x\rceil$$

Let \(\Sigma=\\{0,1\\} .\) Let \(f: \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x)=\|x\| .\) Evaluate \(f(x)\) for each value of \(x\). $$00110011$$

ORD: ASCII \(\rightarrow\) W defined by \(\mathrm{ORD}(\mathrm{c})=\) ordinal number of the character \(c .\)

Rewrite each sum using the summation notation. $$1 \cdot 2+2 \cdot 3+\cdots+11 \cdot 12$$

Let \(A=\left[\begin{array}{lll}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{rrr}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{rrr}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right] .\) Find each. $$B-C$$

Find the number of positive integers \(\leq 3076\) and divisible by: 3 or 4

If 10 points are selected inside an equilateral triangle of unit side, then at least two of them are no more than \(1 / 3\) of a unit apart.

Let \(\Sigma\) denote the English alphabet. Let \(f: \Sigma^{*} \times \Sigma^{*} \rightarrow \Sigma^{*}\) defined by \(f(x, y)=\) \(x y,\) the concatenation of \(x\) and \(y .\) Find \(f(x, y)\) for each pair of words \(x\) and \(y\). combi, natorics

Define the inverse \(g\) of each function \(f\). $$\begin{array}{c|cccc} x & \mathrm{a} & \mathrm{b} & \mathrm{c} & \mathrm{d} \\ \hline f(x) & 4 & 1 & 3 & 2 \end{array}$$

Let \(A\) be an \(m \times n\) matrix, \(B\) a \(p \times q\) matrix, and \(C\) an \(r \times s\) matrix. Under what conditions is each defined? Find the size of each when defined. (Note: \(A^{2}\) means \(A A .\) ) $$B-C$$

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x)=a x+b,\) where \(a, b \in \mathbb{R}\) and \(a \neq 0 .\) Show that \(f\) is surjective; that is, find a real number \(x\) such that \(f(x)=c.\)

Rewrite each sum using the summation notation. $$1(1+2)+2(2+2)+\cdots+5(5+2)$$

Let \(A=\left[\begin{array}{lll}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{rrr}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{rrr}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right] .\) Find each. $$B C$$

Find the number of positive integers \(\leq 3076\) and divisible by: \(3,5,\) or 7

Let \(\Sigma\) denote the English alphabet. Let \(f : \Sigma^{*} \times \Sigma^{*} \rightarrow \Sigma^{*}\) defined by \(f(x, y)=\) \(x y,\) the concatenation of \(x\) and \(y .\) Find \(f(x, y)\) for each pair of words \(x\) and \(y .\) net, work

Define the inverse \(g\) of each function \(f\). $$\begin{array}{c|cccc} x & \mathrm{a} & \mathrm{b} & \mathrm{c} & \mathrm{d} \\ \hline f(x) & \mathrm{b} & \mathrm{c} & \mathrm{d} & \mathrm{a} \end{array}$$

Let \(A\) be an \(m \times n\) matrix, \(B\) a \(p \times q\) matrix, and \(C\) an \(r \times s\) matrix. Under what conditions is each defined? Find the size of each when defined. (Note: \(A^{2}\) means \(A A .\) ) $$B C$$

Determine if each is true or false. $$\sum_{i=m}^{n} i=\sum_{i=m}^{n}(n+m-i)$$

Find the number of positive integers \(\leq 3076\) and divisible by: \(3,5,\) or 6

Prove the following alternate version of the generalized pigeonhole principle: Let \(f: X \rightarrow Y,\) where \(X\) and \(Y\) are finite sets, \(|X|>k \cdot|Y|\) and \(k \in \mathbb{N} .\) Then there is an element \(t \in Y\) such that \(f^{-1}(t)\) contains more than \(k\) elements.

Let \(A=132,33, \ldots, 1261 .\) Let \(f : A \rightarrow\) ASCII defined by \(f(n)=\) character with ordinal number \(n .\) Find \(f(n)\) for each value of \(n .\) $$ 38 $$

Let \(A\) be an \(m \times n\) matrix, \(B\) a \(p \times q\) matrix, and \(C\) an \(r \times s\) matrix. Under what conditions is each defined? Find the size of each when defined. (Note: \(A^{2}\) means \(A A .\) ) $$A^{2}$$

Let \(A=\\{32,33, \ldots, 126\\} .\) Let \(f: A \rightarrow\) ASCII defined by \(f(n)=\) character with ordinal number \(n .\) Find \(f(n)\) for each value of \(n\). $$38$$

Determine if each is true or false. $$\sum_{i=m}^{n} x^{i}=\sum_{i=m}^{n} x^{n+m-i}$$

Let \(A=\left[\begin{array}{lll}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{rrr}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{rrr}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right] .\) Find each. $$A(B+C)$$

Find the number of positive integers \(\leq 3076\) and divisible by: Neither 3 nor 5

Prove that any set \(S\) of three integers contains at least two integers whose sum is even. (Hint: Define a suitable function \(f: S \rightarrow\\{0,1\\}\) and use Exercise \(17 .\) )

Let \(A=132,33, \ldots, 1261 .\) Let \(f : A \rightarrow\) ASCII defined by \(f(n)=\) character with ordinal number \(n .\) Find \(f(n)\) for each value of \(n .\) $$ 64 $$

Determine if the given function is invertible. If it is not invertible, explain why. \(f:\) ASCII \(\rightarrow \mathbf{W}\) defined by \(f(c)=\) ordinal number of the character \(c\).

Let \(A\) be an \(m \times n\) matrix, \(B\) a \(p \times q\) matrix, and \(C\) an \(r \times s\) matrix. Under what conditions is each defined? Find the size of each when defined. (Note: \(A^{2}\) means \(A A .\) ) $$A(B+C)$$