Chapter 4

Let \(A=\\{1,2,4,5\\}, B=\\{2,3,4,6\\},\) and \(C=\\{1,2,3,4\\} .\) Place each of the elements \(1, \ldots, 6\) in the appropriate regions of a three-set Venn diagram.

Let \(A=\\{1,2,\\{1,2\\}, b\\}\) and let \(B=\\{a, b,\\{1,2\\}\\}\). Find the following: (a) \(A \cap B\) (b) \(A \cup B\) (c) \(A \backslash B\) (d) \(B \backslash A\) (e) \(A \triangle B\)

Insert either \(\in\) or \(\subseteq\) in the blanks in the following sentences (in order to produce true sentences). i) \(1 \)__________ \(\longrightarrow\\{3,2,1,\\{a, b\\}\\}\) ii) \(\\{a\\}\) __________ \(\\{a,\\{a, b\\}\\}\) iii) \(\\{a, b\\}\) __________ \(\\{3,2,1,\\{a, b\\}\\}\) iv) \(\\{\\{a, b\\}\\}\)__________ \(-\\{a,\\{a, b\\}\\}\)

What is the power set of \(\emptyset ?\) Hint: if you got the last exercise in the chapter you'd know that this power set has \(2^{0}=1\) element.

One way out of Russell's paradox is to declare that the collection of sets that don't contain themselves as elements is not a set itself. Explain how this circumvents the paradox.

Prove or disprove: $$(A \cap C \subseteq B \cap C) \quad \Longrightarrow \quad A \subseteq B$$.

Try iterating the power set operator. What is \(\mathcal{P}(\mathcal{P}(\emptyset)) ?\) What is \(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))) ?\)

Venn diagrams are usually made using simple closed curves with no further restrictions. Try creating Venn diagrams for 3,4 and 5 sets (in general position) using rectangular simple closed curves.

Provide a counterexample to dispel the notion that a subset must have fewer elements than its superset.

Determine the following cardinalities. (a) \(A=\\{1,2,\\{3,4,5\\}\\} \quad|A|=\) __________ (b) \(B=\\{\\{1,2,3,4,5\\}\\} \quad|B|=\) __________

We have seen that \(A \subseteq B\) corresponds to \(M_{A} \Longrightarrow M_{B}\). What corresponds to the contrapositive statement?

For each positive integer \(n,\) we'll define an interval \(I_{n}\) by $$I_{n}=[-n, 1 / n)$$ Find the union and intersection of all the intervals in this infinite family. $$\begin{array}{l} \bigcup_{n \in \mathbb{N}} I_{n}= \\ \bigcap_{n \in \mathbb{N}} I_{n}= \end{array}$$

Determine two sets \(A\) and \(B\) such that both of the sentences \(A \in B\) and \(A \subseteq B\) are true.

Find the disjunctive normal form of \(A \cap(B \cup C)\).

There is a set \(X\) such that, for all sets \(A,\) we have \(X \Delta A=A .\) What is \(X ?\)

Prove that the set of perfect fourth powers is contained in the set of perfect squares.

Find the disjunctive normal form of \((A \triangle B) \triangle C\).

Find a logical open sentence such that \(\\{0,1,4,9, \ldots\\}\) is its truth set.

The prototypes for the modus ponens and modus tollens argument forms are the following: All men are mortal. Socrates is a man. Therefore Socrates is mortal. and All men are mortal. Zeus is not mortal. Therefore Zeus is not a \(\operatorname{man}\). Illustrate these arguments using Venn diagrams.

Use Venn diagrams to convince yourself of the validity of the following containment statement $$(A \cap B) \cup(C \cap D) \subseteq(A \cup C) \cap(B \cup D)$$ Now prove it!

Prove that \(A \cup(B \cap C)=(A \cup B) \cap(A \cup C)\) by showing that \(A \cup(B \cap C) \subseteq\) \((A \cup B) \cap(A \cup C)\) and \((A \cup B) \cap(A \cup C) \subseteq A \cup(B \cap C)\).

How many 8 element subsets are there in $$ \mathcal{P}(\\{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p\\}) ? $$

Use Venn diagrams to show that the following set equivalence is false. $$(A \cup B) \cap(C \cup D)=(A \cup C) \cap(B \cup D).$$