Chapter 3

Evaluate each sum and product, where \(p\) is a prime and \(I=\\{1,2,3,5\\}.\) $$\sum_{p \leq 10} p$$

The transpose of a matrix \(A=\left(a_{ij}\right)_{m \times n},\) denoted by \(A^{\mathrm{T}},\) is defined as \(A^{\mathrm{T}}=\left(a_{j i}\right)_{n \times m} .\) Find the transpose of each. $$\left[\begin{array}{rrr}{1} & {2} & {3} \\ {2} & {0} & {-1} \\ {-2} & {1} & {0}\end{array}\right]$$

Let \(S=\\{\text { true, falsel. Define a boolean function } f : \mathbb{N} \rightarrow S \text { by } f(n)=\text { true }\) if year \(n\) is a leap year and false otherwise. Find \(f(n)\) for each year \(n .\) $$2020$$

The transpose of a matrix \(A=\left(a_{i j}\right)_{m \times n},\) denoted by \(A^{\mathrm{T}},\) is defined as \(A^{\mathrm{T}}=\left(a_{j i}\right)_{n \times m} \cdot\) Find the transpose of each. $$\left[\begin{array}{rrr} 1 & 2 & 3 \\ 2 & 0 & -1 \\ -2 & 1 & 0 \end{array}\right]$$

Let \(S=\\{\text { true, false }\\} .\) Define a boolean function \(f: \mathbb{N} \rightarrow S\) by \(f(n)=\) true if year \(n\) is a leap year and false otherwise. Find \(f(n)\) for each year \(n\). $$2020$$

Two sets \(A\) and \(B\) are equivalent, denoted by \(A \sim B,\) if there exists a bijection between them. Prove each. \(A \sim A\) (reflexive property)

Evaluate each sum and product, where \(p\) is a prime and \(I=\\{1,2,3,5\\}.\) $$\prod_{p \leq 10} p$$

The transpose of a matrix \(A=\left(a_{ij}\right)_{m \times n},\) denoted by \(A^{\mathrm{T}},\) is defined as \(A^{\mathrm{T}}=\left(a_{j i}\right)_{n \times m} .\) Find the transpose of each. $$\left[\begin{array}{lll}{a} & {b} & {c} \\ {d} & {e} & {f} \\ {f} & {g} & {h}\end{array}\right]$$

Let \(S=\\{\text { true, falsel. Define a boolean function } f : \mathbb{N} \rightarrow S \text { by } f(n)=\text { true }\) if year \(n\) is a leap year and false otherwise. Find \(f(n)\) for each year \(n .\) $$2076$$

The transpose of a matrix \(A=\left(a_{i j}\right)_{m \times n},\) denoted by \(A^{\mathrm{T}},\) is defined as \(A^{\mathrm{T}}=\left(a_{j i}\right)_{n \times m} \cdot\) Find the transpose of each. $$\left[\begin{array}{lll} a & b & c \\ d & e & f \\ f & g & h \end{array}\right]$$

Let \(S=\\{\text { true, false }\\} .\) Define a boolean function \(f: \mathbb{N} \rightarrow S\) by \(f(n)=\) true if year \(n\) is a leap year and false otherwise. Find \(f(n)\) for each year \(n\). $$2076$$

Let \(n \in \mathbb{N}\) . A positive integer \(d\) is a proper factor of \(n\) if \(d\) is a factor of \(n\) and \(d < n .\) For example, the proper factors of 12 are \(1,2,3,4,\) and 6 . Let \(\sigma : N \rightarrow N\) defined by \(\sigma(n)=\) sum of the proper factors of \(n .\) (\sigma is the lowercase Greek letter, sigma.) Compute \(\sigma(n)\) for each value of \(n,\) where \(p\) and \(q\) are distinct primes, IA positive integer \(n\) such that \(\sigma(n)=n\) is a perfect number. $$ 6 $$

Evaluate each sum and product, where \(p\) is a prime and \(I=\\{1,2,3,5\\}.\) $$\begin{aligned} &\prod(3 i-1)\\\ &i \in I \end{aligned}$$

A square matrix \(A\) is symmetric if \(A^{\mathrm{T}}=A .\) What can you say about the elements of a symmetric matrix \(A\) ?

Let \(n \in \mathbb{N}\) . A positive integer \(d\) is a proper factor of \(n\) if \(d\) is a factor of \(n\) and \(d < n .\) For example, the proper factors of 12 are \(1,2,3,4,\) and 6 . Let \(\sigma : N \rightarrow N\) defined by \(\sigma(n)=\) sum of the proper factors of \(n .\) (\sigma is the lowercase Greek letter, sigma.) Compute \(\sigma(n)\) for each value of \(n,\) where \(p\) and \(q\) are distinct primes, IA positive integer \(n\) such that \(\sigma(n)=n\) is a perfect number. $$12$$

Let \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z .\) Prove each. $$f \circ 1_{X}=f$$

Let \(S=\\{\text { true, falsel. Define a boolean function } f : \mathbb{N} \rightarrow S \text { by } f(n)=\text { true }\) if year \(n\) is a leap year and false otherwise. Find \(f(n)\) for each year \(n .\) $$3000$$

A square matrix \(A\) is symmetric if \(A^{\prime \mathrm{T}}=A .\) What can you say about the elements of a symmetric matrix \(A\) ?

Let \(S=\\{\text { true, false }\\} .\) Define a boolean function \(f: \mathbb{N} \rightarrow S\) by \(f(n)=\) true if year \(n\) is a leap year and false otherwise. Find \(f(n)\) for each year \(n\). $$3000$$

Two sets \(A\) and \(B\) are equivalent, denoted by \(A \sim B,\) if there exists a bijection between them. Prove each. If \(A \sim B,\) then \(A \times\\{1\\} \sim B \times\\{2\\}.\)

Let \(A\) be a square matrix. Prove that \(\left(A^{T}\right)^{T}=A.\)

Let \(n \in \mathbb{N}\) . A positive integer \(d\) is a proper factor of \(n\) if \(d\) is a factor of \(n\) and \(d < n .\) For example, the proper factors of 12 are \(1,2,3,4,\) and 6 . Let \(\sigma : N \rightarrow N\) defined by \(\sigma(n)=\) sum of the proper factors of \(n .\) (\sigma is the lowercase Greek letter, sigma.) Compute \(\sigma(n)\) for each value of \(n,\) where \(p\) and \(q\) are distinct primes, IA positive integer \(n\) such that \(\sigma(n)=n\) is a perfect number. $$p q$$

Let \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z .\) Prove each. $$1_{Y} \circ f=f$$

January \(1,2000,\) falls on a Saturday. What day of the week will January \(1,2020,\) be?

Let \(A\) be a square matrix. Prove that \(\left(A^{\mathrm{T}}\right)^{\mathrm{T}}=A\).

January \(1,1990,\) was a Monday. What day of the week was January 1 \(1976 ?\) (Hint: Again, look for leap years.)

Let \(A, B,\) and \(C\) be square matrices of order \(2 .\) Prove each. $$(A+B)^{\mathrm{T}}=A^{\mathrm{T}}+B^{\mathrm{T}}$$

Two sets \(A\) and \(B\) are equivalent, denoted by \(A \sim B,\) if there exists a bijection between them. Prove each. \(\mathbf{Z} \sim \mathbf{O},\) the set of odd integers

If \(f\) and \(g\) are injective, then \(g \circ f\) is injective.

Let \(n \in \mathbb{N}\) . A positive integer \(d\) is a proper factor of \(n\) if \(d\) is a factor of \(n\) and \(d < n .\) For example, the proper factors of 12 are \(1,2,3,4,\) and 6 . Let \(\sigma : N \rightarrow N\) defined by \(\sigma(n)=\) sum of the proper factors of \(n .\) (\sigma is the lowercase Greek letter, sigma.) Compute \(\sigma(n)\) for each value of \(n,\) where \(p\) and \(q\) are distinct primes, IA positive integer \(n\) such that \(\sigma(n)=n\) is a perfect number. $$p^{2}$$

Prove. A bijection exists between any two closed intervals \([a, b]\) and \([c, d],\) where \(a< b\) and \(c< d\) . (Hint: Find a suitable function that works.)

Using the functions \(f(x)=2 x+3\) and \(g(x)=x^{2}-1,\) find the following. $$(f+g)(-3)$$

Let \(A, B,\) and \(C\) be square matrices of order \(2 .\) Prove each. $$(A B)^{\mathrm{T}}=B^{\mathrm{T}} A^{\mathrm{T}}$$

If \(f\) and \(g\) are surjective, then \(g \circ f\) is surjective.

Each day of the week, beginning with Sunday, can be identified by a code \(x,\) where \(0 \leq x \leq 6 .\) January 1 of any year \(y\) can be determined using the following formula^{** } $$ x \equiv\left(y+\left\lfloor\frac{y-1}{4}\right\rfloor-\left\lfloor\frac{y-1}{100}\right\rfloor+\left\lfloor\frac{y-1}{400}\right\rfloor\right) \bmod 7 $$ Using this formula determine the first day in each year. $$2000$$

Prove each. A bijection exists between any two closed intervals \([a, b]\) and \([c, d]\) where \(a

Prove. The set of odd positive integers is countably infinite.

Evaluate each sum and product, where \(p\) is a prime and \(I=\\{1,2,3,5\\}.\) $$\sum_{p \leq 25} 1$$

Using the functions \(f(x)=2 x+3\) and \(g(x)=x^{2}-1,\) find the following. $$(f g)(2)$$

Let \(A, B,\) and \(C\) be square matrices of order \(2 .\) Prove each. $$\left(A A^{\mathrm{T}}\right)^{\mathrm{T}}=A A^{\mathrm{T}}$$

If \(f\) and \(g\) are bijective, then \(g\) of is bijective.

Prove each. The set of odd positive integers is countably infinite.

If \(f\) and \(g\) are bijective, then \(g \circ f\) is bijective.

Each day of the week, beginning with Sunday, can be identified by a code \(x,\) where \(0 \leq x \leq 6 .\) January 1 of any year \(y\) can be determined using the following formula^{** } $$ x \equiv\left(y+\left\lfloor\frac{y-1}{4}\right\rfloor-\left\lfloor\frac{y-1}{100}\right\rfloor+\left\lfloor\frac{y-1}{400}\right\rfloor\right) \bmod 7 $$ Using this formula determine the first day in each year. $$2020$$

Prove. The set of integers is countably infinite.

Evaluate each sum and product, where \(p\) is a prime and \(I=\\{1,2,3,5\\}.\) $$\begin{aligned} &\prod(i+2 j)\\\ &i, j \in I\\\ &i

Using the functions \(f(x)=2 x+3\) and \(g(x)=x^{2}-1,\) find the following. $$(f+g)(x)$$

Let \(A, B,\) and \(C\) be square matrices of order \(2 .\) Prove each. $$(A B C)^{\mathrm{T}}=C^{\mathrm{T}} B^{\mathrm{T}} A^{\mathrm{T}}$$

The identity function \(1_{X}\) is bijective.

Each day of the week, beginning with Sunday, can be identified by a code \(x,\) where \(0 \leq x \leq 6 .\) January 1 of any year \(y\) can be determined using the following formula^{** } $$ x \equiv\left(y+\left\lfloor\frac{y-1}{4}\right\rfloor-\left\lfloor\frac{y-1}{100}\right\rfloor+\left\lfloor\frac{y-1}{400}\right\rfloor\right) \bmod 7 $$ Using this formula determine the first day in each year. $$2076$$