91Ó°ÊÓ

To prove the following set equalities, it may be necessary to use some of the properties of positive and negative real numbers. For example, it may be necessary to use the facts that: \- The product of two real numbers is positive if and only if the two real numbers are either both positive or both negative. \- The product of two real numbers is negative if and only if one of the two numbers is positive and the other is negative. For example, if \(x(x-2)<0,\) then we can conclude that either \((1) x<0\) and \(x-2>0\) or \((2) x>0\) and \(x-2<0 .\) However, in the first case, we must have \(x<0\) and \(x>2\), and this is impossible. Therefore, we conclude that \(x>0\) and \(x-2<0,\) which means that \(0

Use the definitions of set intersection, set union, and set difference to write useful negations of these definitions. That is, complete each of the following sentences (a) \(x \notin A \cap B\) if and only if \(\ldots\) (b) \(x \notin A \cup B\) if and only if \(\ldots\) (c) \(x \notin A-B\) if and only if \(\ldots\)

Let \(A, B,\) and \(C\) be subsets of some universal set \(U\) (a) Draw two general Venn diagrams for the sets \(A, B,\) and \(C .\) On one, shade the region that represents \(A-(B-C),\) and on the other, shade the region that represents \((A-B)-C .\) Based on the Venn diagrams, make a conjecture about the relationship between the sets \(A-(B-C)\) and \((A-B)-C .\) (Are the two sets equal? If not, is one of the sets a subset of the other set?) (b) Prove the conjecture from Exercise (7a).

Let \(A\) and \(B\) be subsets of some universal set \(U\). (a) Prove that \(A\) and \(B-A\) are disjoint sets. (b) Prove that \(A \cup B=A \cup(B-A)\).

Is the following proposition true or false? Justify your conclusion with a proof or a counterexample. For all sets \(A\) and \(B\) that are subsets of some universal set \(U,\) the sets \(A \cap B\) and \(A-B\) are disjoint.

Give an example of an indexed family of sets \(\left\\{A_{n} \mid n \in \mathbb{N}\right\\}\) such all three of the following conditions are true: (i) For each \(m \in \mathbb{N}, A_{m} \subseteq(0,1) ;\) (ii) For each \(j, k \in \mathbb{N}\), if \(j \neq k\), then \(A_{j} \cap A_{k} \neq \emptyset\); and (iii) \(\bigcap_{k \in \mathbb{N}} A_{k}=\emptyset\).

Let \(A, B,\) and \(C\) be subsets of some universal set \(U\). For each of the following, draw a general Venn diagram for the three sets and then shade the indicated region. (a) \(A \cap B\) (b) \(A \cap C\) (c) \((A \cap B) \cup(A \cap C)\) (d) \(B \cup C\) (e) \(A \cap(B \cup C)\) (f) \((A \cap B)-C\)

See the instructions for Exercise (19) on page 100 from Section 3.1 . (a) If \(A, B,\) and \(C\) are subsets of some universal set \(U,\) then \(A-(B-C)=\) \(A-(B \cup C)\) Proof. $$ \begin{aligned} A-(B-C) &=(A-B)-(A-C) \\ &=\left(A \cap B^{c}\right) \cap\left(A \cap C^{c}\right) \\ &=A \cap\left(B^{c} \cap C^{c}\right) \\ &=A \cap(B \cup C)^{c} \\ &=A-(B \cup C) \end{aligned} $$ (b) If \(A, B,\) and \(C\) are subsets of some universal set \(U,\) then \(A-(B \cup C)=\) \((A-B) \cap(A-C)\) \mathrm{\\{} P r o o f . ~ W e ~ f i r s t ~ w r i t e ~ \(A-(B \cup C)=A \cap(B \cup C)^{c}\) and then use one of De Morgan's Laws to obtain $$ A-(B \cup C)=A \cap\left(B^{c} \cap C^{c}\right) $$ We now use the fact that \(A=A \cap A\) and obtain \(A-(B \cup C)=A \cap A \cap B^{c} \cap C^{c}=\left(A \cap B^{c}\right) \cap\left(A \cap C^{c}\right)=\) \((A-B) \cap(A-C)\)

We can extend the idea of consecutive integers (See Exercise (10) in Section 3.5) to represent four consecutive integers as \(m, m+1, m+2,\) and \(m+3,\) where \(m\) is an integer. There are other ways to represent four consecutive integers. For example, if \(k \in \mathbb{Z}\), then \(k-1, k, k+1,\) and \(k+2\) are four consecutive integers. (a) Prove that for each \(n \in \mathbb{Z}, n\) is the sum of four consecutive integers if and only if \(n \equiv 2(\bmod 4)\). (b) Use set builder notation or the roster method to specify the set of integers that are the sum of four consecutive integers. (c) Specify the set of all natural numbers that can be written as the sum of four consecutive natural numbers. (d) Prove that for each \(n \in \mathbb{Z}, n\) is the sum of eight consecutive integers if and only if \(n \equiv 4(\bmod 8)\) (e) Use set builder notation or the roster method to specify the set of integers that are the sum of eight consecutive integers. (f) Specify the set of all natural numbers can be written as the sum of eight consecutive natural numbers.

Let \(A, B,\) and \(C\) be subsets of a universal set \(U .\) Are the following propositions true or false? Justify your conclusions. (a) If \(A \cap C \subseteq B \cap C,\) then \(A \subseteq B\). (b) If \(A \cup C \subseteq B \cup C,\) then \(A \subseteq B\). (c) If \(A \cup C=B \cup C,\) then \(A=B\). (d) If \(A \cap C=B \cup C,\) then \(A=B\). (e) If \(A \cup C=B \cup C\) and \(A \cap C=B \cap C,\) then \(A=B\).