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

Give an example of a function from N to N that is

1. One-to-one but not onto.
2. Onto but not one-to-one
3. Both onto and one-to-one(but different from the identity function).
4. Neither one-to-one nor onto.

For which real numbers xand yis it true that(x+y) =

[x] + [y]?

Show that if A, B and C are sets such that $\left|\mathit{A}\right|\le \left|\mathit{B}\right|\text{and}\left|\mathit{B}\right|\le \left|\mathit{C}\right|\text{, then}\left|\mathit{A}\right|\le \left|\mathit{C}\right|$.

Prove the first associative law from Table 1 by showing that if A, B, and C are sets, then A∪(B∪C) = (A∪B)∪C.

A factory makes custom sports cars at an increasing rate. In the first month only one car is made, in the second month two cars are made, and so on, with cars made in the nth month.

1. Set up a recurrence relation for the number of cars produced in the firstmonths by this factory.
2. How many cars are produced in the first year?
3. Find an explicit formula for the number of cars produced in the first months by this factory.

Find the power set of each of these sets, where a and b are distinct elements?

(a) \(\left\{ {\bf{a}} \right\}\)

(b) \(\left\{ {{\bf{a,b}}} \right\}\)

(c) \(\left\{ {\phi \left\{ \phi \right\}} \right\}\)

Give an explicit formula for a function from the set of integers to the set of positive integers that is

1. One-to-one, but not onto.
2. Onto, but not one-to-one
3. One-to-one and onto
4. Neither One-to-one nor onto

Let A be an invertible matrix. Show that $\mathbf{(}{\mathit{A}}^{\mathbf{n}}{\mathbf{)}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\mathbf{(}{\mathit{A}}^{\mathbf{-}\mathbf{1}}{\mathbf{)}}^{\mathbf{n}}$ whenever n is a positive integer.

For which real numbers xand yis it true that(x+y) =

[x] + [y]?

Let A be a matrix. Show that the matrix is symmetric. [Hint: Show that this matrix $\mathit{A}{\mathit{A}}^{\mathbf{t}}$ equals its transpose with the help of Exercise 17b.]