91Ó°ÊÓ

In Section 1.1, we studied some of the closure properties of the standard number systems. (See page 10.) We can extend this idea to other sets of numbers. So we say that: A set \(A\) of numbers is closed under addition provided that whenever \(x\) and \(y\) are in the set \(A, x+y\) is in the set \(A\). \- A set \(A\) of numbers is closed under multiplication provided that whenever \(x\) and \(y\) are are in the set \(A, x \cdot y\) is in the set \(A\). \- A set \(A\) of numbers is closed under subtraction provided that whenever \(x\) and \(y\) are are in the set \(A, x-y\) is in the set \(A\). For each of the following sets, make a conjecture about whether or not it is closed under addition and whether or not it is closed under multiplication. In some cases, you may be able to find a counterexample that will prove the set is not closed under one of these operations. (a) The set of all odd natural numbers (b) The set of all even integers (c) \(A=\\{1,4,7,10,13, \ldots\\}\) (d) \(B=\\{\ldots,-6,-3,0,3,6,9, \ldots\\}\) bers \(\quad\) (e) \(C=\\{3 n+1 \mid n \in \mathbb{Z}\\}\) (f) \(D=\left\\{\frac{1}{2^{n}} \mid n \in \mathbb{N}\right\\}\)

Suppose each of the following statements is true. \- Laura is in the seventh grade. \- Laura got an \(\mathrm{A}\) on the mathematics test or Sarah got an \(\mathrm{A}\) on the mathematics test. \- If Sarah got an \(\mathrm{A}\) on the mathematics test, then Laura is not in the seventh grade. If possible, determine the truth value of each of the following statements. Carefully explain your reasoning. (a) Laura got an \(\mathrm{A}\) on the mathematics test. (b) Sarah got an \(\mathrm{A}\) on the mathematics test. (c) Either Laura or Sarah did not get an \(\mathrm{A}\) on the mathematics test.

In calculus, we define a function \(f\) with domain \(\mathbb{R}\) to be strictly increasing provided that for all real numbers \(x\) and \(y, f(x)

Let \(a, b,\) and \(c\) be integers. Consider the following conditional statement: If \(a\) divides \(b c,\) then \(a\) divides \(b\) or \(a\) divides \(c\). Which of the following statements have the same meaning as this conditional statement and which ones are negations of this conditional statement? The note for Exercise (10) also applies to this exercise. (a) If \(a\) divides \(b\) or \(a\) divides \(c,\) then \(a\) divides \(b c\). (b) If \(a\) does not divide \(b\) or \(a\) does not divide \(c,\) then \(a\) does not divide \(b c\). (c) \(a\) divides \(b c, a\) does not divide \(b\), and \(a\) does not divide \(c\). (d) If \(a\) does not divide \(b\) and \(a\) does not divide \(c,\) then \(a\) does not divide \(b c\) (e) \(a\) does not divide \(b c\) or \(a\) divides \(b\) or \(a\) divides \(c\). (f) If \(a\) divides \(b c\) and \(a\) does not divide \(c,\) then \(a\) divides \(b\). (g) If \(a\) divides \(b c\) or \(a\) does not divide \(b\), then \(a\) divides \(c\).

For statements \(P, Q,\) and \(R\) : (a) Show that \([(P \rightarrow Q) \wedge P] \rightarrow Q\) is a tautology. Note: In symbolic logic, this is an important logical argument form called modus ponens. (b) Show that \([(P \rightarrow Q) \wedge(Q \rightarrow R)] \rightarrow(P \rightarrow R)\) is a tautology. Note: In symbolic logic, this is an important logical argument form called Syllogism.

Prime Numbers. The following definition of a prime number is very important in many areas of mathematics. We will use this definition at various places in the text. It is introduced now as an example of how to work with a definition in mathematics. Definition. A natural number \(p\) is a prime number provided that it is greater than 1 and the only natural numbers that are factors of \(p\) are 1 and \(p .\) A natural number other than 1 that is not a prime number is a composite number. The number 1 is neither prime nor composite. Using the definition of a prime number, we see that \(2,3,5,\) and 7 are prime numbers. Also, 4 is a composite number since \(4=2 \cdot 2 ; 10\) is a composite number since \(10=2 \cdot 5 ;\) and 60 is a composite number since \(60=4 \cdot 15\). (a) Give examples of four natural numbers other than \(2,3,5,\) and 7 that are prime numbers. (b) Explain why a natural number \(p\) that is greater than 1 is a prime number provided that For all \(d \in \mathbb{N},\) if \(d\) is a factor of \(p,\) then \(d=1\) or \(d=p\). (c) Give examples of four natural numbers that are composite numbers and explain why they are composite numbers. (d) Write a useful description of what it means to say that a natural number is a composite number (other than saying that it is not prime).

Suppose we are trying to prove the following for integers \(x\) and \(y\) : If \(x \cdot y\) is even, then \(x\) is even or \(y\) is even. We notice that we can write this statement in the following symbolic form: $$ P \rightarrow(Q \vee R) $$ where \(P\) is " \(x \cdot y\) is even," \(Q\) is " \(x\) is even," and \(R\) is " \(y\) is even." (a) Write the symbolic form of the contrapositive of \(P \rightarrow(Q \vee R)\). Then use one of De Morgan's Laws (Theorem 2.5 ) to rewrite the hypothesis of this conditional statement. (b) Use the result from Part (13a) to explain why the given statement is logically equivalent to the following statement: If \(x\) is odd and \(y\) is odd, then \(x \cdot y\) is odd. The two statements in this activity are logically equivalent. We now have the choice of proving either of these statements. If we prove one, we prove the other, or if we show one is false, the other is also false. The second statement is Theorem \(1.8,\) which was proven in Section 1.2 .