diff --git a/chapter_4/exercises.md b/chapter_4/exercises.md index 07d3b9c..d74fafa 100644 --- a/chapter_4/exercises.md +++ b/chapter_4/exercises.md @@ -637,3 +637,182 @@ b. \(c\) $10k^2 + 10k + 6$ (d) $5n^2 + 7$ is even + +--- + +**Exercise Set 4.2** + +Page 204 + +Prove the statements in 1-11. In each case use only the definitions of the terms +and the Assumptions listed on page 161, not any previously established +properties of odd and even integers. Follow the directions given in this section +for writing proofs of universal statements. + +1. For every integer $n$, if $n$ is odd then $3n + 5$ is even. + +2. For ever integer $m$, if $m$ is even then $3m + 5$ is odd. + +3. For every integer $n$, $2n - 1$ is odd. + +4. **Theorem 4.2.2:** The difference of any even integer minus any odd integer + is odd. + +5. If $a$ and $b$ are any odd integers, then $a^2 + b^2$ is even. + +6. If $k$ is any odd integer and $m$ is any even integer, then $k^2 + m^2$ is + odd. + +7. The difference between the squares of any two consecutive integers is odd. + +8. For any integers $m$ and $n$, if $m$ is even and $n$ is odd then $5m + 3n$ is + odd. + +9. If an integer greater than $4$ is a perfect square, then the immediately + preceding integer is not prime. + +10. If $n$ is any even integer, then $(-1)^n = 1$. + +11. If $n$ is any odd integer, then $(-1)^n = -1$. + +Prove that the statements in 12-14 are false. + +12. There exists an integer $m \geq 3$ such that $m^2 - 1$ is prime. + +13. There exists an integer $n$ such that $6n^2 + 27$ is prime. + +14. There exists an integer $k \geq 4$ such that $2k^2 - 5k + 2$ is prime. + +Find the mistakes in the "proofs" shown in 15-19. + +15. + +**Theorem:** For every integer $k$, if $k > 0$ then $k^2 + 2k + 1$ is composite. + +**"Proof:** For $k = 2$, $k > 0$ and $k^2 + 2k + 1 = 2^2 + 2 \cdot 2 + 1 = 9$. +And since $9 = 3 \cdot 3$, then $9$ is composite. Hence the theorem is true." + +16. + +**Theorem:** The difference between any odd integer and any even integer is odd. + +**"Proof:** Suppose $n$ is any odd integer, and $m$ is any even integer. By +definition of odd, $n = 2k + 1$ where $k$ is an integer, and by definition of +even, $m = 2k$ where $k$ is an integer. Then + +$$ n - m = (2k + 1) - 2k = 1 $$ + +and $1$ is odd. Therefore, the difference between any odd integer and any even +integer is odd." + +17. + +**Theorem:** For every integer $k$, if $k > 0$, then $k^2 + 2k + 1$ is +composite. + +**"Proof:** Suppose $k$ is any integer such that $k > 0$. If $k^2 + 2k + 1$ is +composite, then $k^2 + 2k + 1 = rs$ for some integers $r$ and $s$ such that + +$$ 1 < r < k^2 + 2k + 1 $$ + +and + +$$ 1 < s < k^2 + 2k + 1 $$ + +Since + +$$ k^2 + 2k + 1 = rs $$ + +and both $r$ and $s$ are strictly between $1$ and $k^2 + 2k + 1$, then +$k^2 + 2k + 1$ is not prime. Hence $k^2 + 2k + 1$ is composite as was to be +shown." + +18. + +**Theorem:** The product of any even integer and any odd integer is even. + +**"Proof:** Suppose $m$ is any even integer and $n$ is any odd integer. If +$m \cdot n$ is even, then by definition of even there exists an integer $r$ such +that $m \cdot n = 2r$. Also since $m$ is even, there exists an integer $p$ such +that $m = 2p$, and since $n$ is odd there exists an integer $q$ such that +$n = 2q + 1$. Thus + +$$ mn = (2p)(2q + 1) = 2r $$ + +where $r$ is an integer. By definition of even, then, $m \cdot n$ is even, as +was to be shown." + +19. + +**Theorem:** The sum of any two even integers equals $4k$ for some integer $k$. + +**"Proof:** Suppose $m$ and $n$ are any two even integers. By definition of +even, $m = 2k$ for some integer $k$ and $n = 2k$ for some integer $k$. By +substitution, + +$$ m + n = 2k + 2k = 4k $$ + +That is what was to be shown." + +In 20-38 determine whether the statement is true or false. Justify your answer +with a proof or a counterexample, as appropriate. In each case use only the +definitions of the terms and the Assumptions listed on page 161, not any +previously established properties. + +20. The product of any two odd integers is odd. + +21. The negative of any odd integer is odd. + +22. For all integers $a$ and $b$, $4a + 5b + 3$ is even. + +23. The product of any even integer and any integer is even. + +24. If a sum of two integers is even, then one of the summands is even. (In the + expression $a + b$, $a$ and $b$ are called **summands**.) + +25. The difference of any two even integers is even. + +26. For all integers $a$, $b$, and $c$, if $a$, $b$, and $c$ are consecutive, + then $a + b + c$ is even. + +27. The difference of any two odd integers is even. + +28. For all integers $n$ and $m$, if $n - m$ is even then $n^3 - m^3$ is even. + +29. For every integer $n$, if $n$ is prime then $(-1)^n = -1$. + +30. For every integer $m$, if $m > 2$ then $m^2 - 4$ is composite. + +31. For every integer $n$, $n^2 - n + 11$ is a prime number. + +32. For every integer $n$, $4(n^2 + n + 1) - 3n^2$ is a perfect square. + +33.. Every positive integer can be expressed as a sum of three or fewer perfect +squares. + +34. (Two integers are **consecutive** if, and only if, one is one more than the + other.) Any product of four consecutive integers is one less than a perfect + square. + +35. If $m$ and $n$ are any positive integers and $mn$ is a perfect square, then + $m$ and $n$ are perfect squares. + +36. The difference of the squares of any two consecutive integers is odd. + +37. For all nonnegative real numbers $a$ and $b$, + $\sqrt{ab} = \sqrt{a}\sqrt{b}$. (Note that if $x$ is a nonnegative real + number, then there is a unique nonnegative real number $y$, denoted + $\sqrt{x}$, such that $y^2 = x$.) + +38. For all nonnegative real numbers $a$ and $b$, + +$$ \sqrt{a + b} = \sqrt{a} + \sqrt{b} $$ + +39. Suppose that integers $m$ and $n$ are perfect squares. Then + $m + n + 2\sqrt{mn}$ is also a perfect square. Why? + +40. If $p$ is a prime number, must $2^p - 1$ also be prime? Prove or give a + counterexample. + +41. If $n$ is a nonnegative integer, must $2^{2n} + 1$ be prime? Prove or give a + counterexample. diff --git a/chapter_4/notes.md b/chapter_4/notes.md index d7afd77..499bb45 100644 --- a/chapter_4/notes.md +++ b/chapter_4/notes.md @@ -103,3 +103,11 @@ $$ m + n = 2r \quad \text{where } t \text{ is an integer} $$ It follows by definition of even that $m + n$ is even. _[This is what we needed to show.]_ + +--- + +Page 196 + +Personal Note: The entirety of 4.2 is extremely helpful in breaking down in +exactly how to write proofs (for beginners). I'd advise revisiting this entire +section frequently. diff --git a/chapter_4/test_yourself.md b/chapter_4/test_yourself.md index 13be4a0..48a010f 100644 --- a/chapter_4/test_yourself.md +++ b/chapter_4/test_yourself.md @@ -32,3 +32,33 @@ property. $x$ is a particular but arbitrarily chosen element of the set $D$ that makes the hypothesis $P(x)$ true; $x$ makes the conclusion $Q(x)$ true. + +--- + +**Test Yourself** + +Page 204 + +1. The meaning of every variable used in a proof should be explained with + ______. + +2. Proofs should be written in sentences that are ______ and ______. + +3. Every assertion in a proof should be supported by a ______. + +4. The following are some useful "little words and phrases" that clarify the + reasoning in a proof: + + ______, ______, ______, ______, and ______. + +5. A new thought or fact that does not follow as an immediate consequence of the + preceding statement can be introduced by writing ______, ______, ______, + ______, or ______. + +6. To introduce a new variable that is defined in terms of previous variables, + use the word ______. + +7. Displaying equations and inequalities increases the ______ of a proof. + +8. Some proof-writing mistakes are ______, ______, ______, ______, ______, + ______, and ______.