Chapter 3

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 \(n\). $$64$$

Determine if each function \(f: A \rightarrow B\) is bijective. $$f(x)=|x|, A=B=\mathbb{R}$$

Sums of the form \(S=\sum_{i=m+1}^{n}\left(a_{i}-a_{i-1}\right)\) are telescoping sums. Show that \(S=a_{n}-a_{m} .\)

Determine if the given function is invertible. If it is not invertible, explain why. $$f: \mathbf{W} \rightarrow \mathbf{W} \text { defined by } f(n)=n(\bmod 5)$$

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. $$A B-A C$$

Using the pigeonhole principle, prove that the cardinality of a finite set is unique.

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 \(n .\) $$ 90 $$

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 \(A A .\) ) $$A B-A C$$

Determine if each is true or false. Sums of the form \(S=\sum_{i=m+1}^{n}\left(a_{i}-a_{i-1}\right)\) are telescoping sums. Show that \(S=a_{n}-a_{m} . \quad i=m+1.\)

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 \(n\). $$90$$

Determine if the given function is invertible. If it is not invertible, explain why. $$f: \Sigma^{*} \rightarrow \Sigma^{*} \text { defined by } f(w)=\text { awa, where } \Sigma=\\{\mathbf{a}, \mathbf{b}, \mathbf{c}\\}$$.

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. $$A B+C$$

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 \(A A .\) ) $$A B+C$$

Determine if each is true or false. Using Exercise 19 and the identity \(\frac{1}{i(i+1)}=\frac{1}{i}-\frac{1}{i+1},\) derive a formula for \(\sum_{i=1}^{n} \frac{1}{i(i+1)}.\)

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 \(n\). $$123$$

Using Exercise 19 and the identity \((i+1)^{2}-i^{2}=2 i+1,\) find a formula for $$\sum_{i=1}^{n} i$$

Determine if the given function is invertible. If it is not invertible, explain why. \(f: S \rightarrow \mathbb{N}\) defined by \(f(x)=\) decimal value of \(x,\) where \(S\) is the set of binary representations of positive integers with no leading zeros.

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. $$A(B C)$$

Let \(g: \mathrm{ASCII} \rightarrow A\) defined by \(g(c)=n,\) where \(A=\\{32,33, \ldots, 126\\}\) and \(n\) denotes the ordinal number of the character \(c .\) Find \(g(c)\) for each character \(c .\) $$'+'$$

Determine if each is true or false. Using Exercise 19 and the identity \((i+1)^{2}-i^{2}=2 i+1,\) find a formula for \(\sum_{i=1}^{n} i\)

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 \(A A .\) ) $$A(B C)$$

Determine if the given function is invertible. If it is not invertible, explain why. \(f: \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x)=\) decimal value of \(x,\) where \(\Sigma=\\{0,1\\}\).

Let \(g: \mathrm{ASCII} \rightarrow A\) defined by \(g(c)=n,\) where \(A=\\{32,33, \ldots, 126\\}\) and \(n\) denotes the ordinal number of the character \(c .\) Find \(g(c)\) for each character \(c .\) $$'<'$$

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll} 1 & \text { if } i=j \\ 0 & \text { otherwise } \end{array}\right. $$ $$\sum_{i=1}^{5} \sum_{j=1}^{6}(2 i+3 j)$$

Determine if the functions are bijective. If they are not bijective, explain why. \(f: \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x)=\) decimal value of \(x,\) where \(\Sigma=\\{0,1\\}.\)

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll}{1} & {\text { if } i=j} \\ {0} & {\text { otherwise }}\end{array}\right. $$ \(\left[\delta_{j} \text { is called Kronecker's delta, after the German mathematician Leopold }\right.\) Kronecker \((1823-1891) . ]\) $$ \sum_{i=1}^{3} \sum_{j=1}^{i}(j+3) $$

Let \(g : \mathrm{ASCII} \rightarrow\) \(A\) defined by \(g(c)=n,\) where \(A=\\{32,33, \ldots, 126\\}\) and \(n\) denotes the ordinal number of the character \(c .\) Find \(g(c)\) for each character \(c .\) $$ ^{\prime}{z}^{\prime} $$

Determine if the given function is invertible. If it is not invertible, explain why. Let \(f: \Sigma^{n} \rightarrow \mathbf{W}\) defined by \(f(x)=\sum_{i=1}^{n} x_{i},\) where \(\Sigma^{n}\) denotes the set of words of length \(n\) over \(\Sigma=\\{0,1,2\\}\) and \(x=x_{1} x_{2} \cdots x_{n} .[f(x)\) is the weight of \(x ; \text { for example, } f(10211)=5 .]\).

Let \(U=\\{\mathrm{a}, \ldots, \mathrm{g}\\} .\) Define the characteristic function \(h\) of each set. $$\\{\mathrm{a}, \mathrm{c}, \mathrm{d}, \mathrm{f}\\}$$

Determine if the functions in are bijective. If they are not bijective, explain why. \(f : \Sigma^{*} \times \Sigma^{*} \rightarrow \Sigma^{*}\) defined by \(f(x, y)=x y,\) where \(\Sigma\) denotes the English alphabet.

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll}{1} & {\text { if } i=j} \\ {0} & {\text { otherwise }}\end{array}\right. $$ \(\left[\delta_{j} \text { is called Kronecker's delta, after the German mathematician Leopold }\right.\) Kronecker \((1823-1891) . ]\) $$ \sum_{j=1}^{6} \sum_{i=1}^{5}(2 i+3 i) $$

Mark each sentence as true or false. Assume the composites and inverses are defined: The composition of two injections is injective.

Let \(g : \mathrm{ASCII} \rightarrow\) \(A\) defined by \(g(c)=n,\) where \(A=\\{32,33, \ldots, 126\\}\) and \(n\) denotes the ordinal number of the character \(c .\) Find \(g(c)\) for each character \(c .\) $$ ^{\prime}\left\\{^{\prime}\right. $$

Determine if the functions are bijective. If they are not bijective, explain why. \(f: \Sigma^{*} \times \Sigma^{*} \rightarrow \Sigma^{*}\) defined by \(f(x, y)=x y,\) where \(\Sigma\) denotes the English alphabet.

Let \(U=\\{\mathrm{a}, \ldots, \mathrm{g}\\} .\) Define the characteristic function \(h\) of each set. $$\\{\mathrm{a}, \mathrm{e}, \mathrm{g}\\}$$

Determine if the functions are bijective. If they are not bijective, explain why. \(g: \Sigma^{*} \rightarrow \Sigma^{*}\) defined by \(g(w)=a w a,\) where \(\Sigma=\\{a, b, c\\}.\)

Mark each sentence as true or false. Assume the composites and inverses are defined: The composition of two surjections is surjective.

Let \(f: \mathbf{Z} \times \mathbf{Z} \rightarrow \mathbf{Z}\) defined by \(f(x, y)=2 x+3 y-6 x y .\) Compute the following. $$f(2,3)$$

Let \(U = \\{a,\ldots,g | .\text { Define the characteristic function } h \text { of each set. }\) $$\\{\mathrm{b}, \mathrm{c}, \mathrm{g}\\}$$

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll} 1 & \text { if } i=j \\ 0 & \text { otherwise } \end{array}\right. $$ $$\sum_{i=1}^{6} \sum_{j=1}^{5}\left(i^{2}-i\right)$$

Let \(U=\\{\mathrm{a}, \ldots, \mathrm{g}\\} .\) Define the characteristic function \(h\) of each set. $$\\{\mathrm{b}, \mathrm{c}, \mathrm{g}\\}$$

Determine if the functions are bijective. If they are not bijective, explain why. \(f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \times \mathbb{R}\) defined by \(f(x, y)=(x,-y).\)

Mark each sentence as true or false. Assume the composites and inverses are defined: The composition of two bijections is a bijection.

Let \(f: \mathbf{Z} \times \mathbf{Z} \rightarrow \mathbf{Z}\) defined by \(f(x, y)=2 x+3 y-6 x y .\) Compute the following. $$f(-3,0)$$

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll} 1 & \text { if } i=j \\ 0 & \text { otherwise } \end{array}\right. $$ $$\sum_{i=1}^{5} \sum_{j=1}^{6}\left(i^{2}-j+1\right)$$

Let \(U=\\{\mathrm{a}, \ldots, \mathrm{g}\\} .\) Define the characteristic function \(h\) of each set. $$\\{\mathrm{a}, \mathrm{c}, \mathrm{d}, \mathrm{f}, \mathrm{g}\\}$$

Determine if the functions are bijective. If they are not bijective, explain why. The ORD function on ASCII.

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll}{1} & {\text { if } i=j} \\ {0} & {\text { otherwise }}\end{array}\right. $$ \(\left[\delta_{j} \text { is called Kronecker's delta, after the German mathematician Leopold }\right.\) Kronecker \((1823-1891) . ]\) $$ \sum_{j=1}^{6} \sum_{i=1}^{5}\left(i^{2}-j+1\right) $$

Let \(A, B,\) and \(C\) be any \(m \times n\) matrices, \(O\) the \(m \times n\) zero matrix, and \(c\) and \(d\) any real numbers. Prove each (see Theorem 3.12). $$A+B=B+A$$

Let \(f: \mathbf{Z} \times \mathbf{Z} \rightarrow \mathbf{Z}\) defined by \(f(x, y)=2 x+3 y-6 x y .\) Compute the following. $$f(-2,3)$$