91Ó°ÊÓ

Consider the statement "If Oscar eats Chinese food, then he drinks milk." (a) Write the converse of the statement. (b) Write the contrapositive of the statement. (c) Is it possible for the contrapositive to be false? If it was, what would that tell you? (d) Suppose the original statement is true, and that Oscar drinks milk. Can you conclude anything (about his eating Chinese food)? Explain. (e) Suppose the original statement is true, and that Oscar does not drink milk. Can you conclude anything (about his eating Chinese food)? Explain.

You have discovered an old paper on graph theory that discusses the viscosity of a graph (which for all you know, is something completely made up by the author). A theorem in the paper claims that "if a graph satisfies condition \((V)\), then the graph is viscous." Which of the following are equivalent ways of stating this claim? Which are equivalent to the converse of the claim? (a) A graph is viscous only if it satisfies condition (V). (b) A graph is viscous if it satisfies condition (V). (c) For a graph to be viscous, it is necessary that it satisfies condition(V). (d) For a graph to be viscous, it is sufficient for it to satisfy condition(V). (e) Satisfying condition (V) is a sufficient condition for a graph to be viscous. (f) Satisfying condition ( \(\mathrm{V}\) ) is a necessary condition for a graph to be viscous. (g) Every viscous graph satisfies condition (V). (h) Only viscous graphs satisfy condition (V).

Find an example of sets \(A\) and \(B\) such that \(A \cap B=\\{3,5\\}\) and \(A \cup B=\\{2,3,5,7,8\\}\).

Let \(P(x)\) be the predicate, \(" 3 x+1\) is even." (a) Is \(P(5)\) true or false? (b) What, if anything, can you conclude about \(\exists x P(x)\) from the truth value of \(P(5) ?\) (c) What, if anything, can you conclude about \(\forall x P(x)\) from the truth value of \(P(5) ?\)

Let \(A=\\{1,2,3, \ldots, 10\\} .\) Consider the function \(f: \mathcal{P}(A) \rightarrow \mathbb{N}\) given by \(f(B)=|B|\). That is, \(f\) takes a subset of \(A\) as an input and outputs the cardinality of that set. (a) Is \(f\) injective? Prove your answer. (b) Is \(f\) surjective? Prove your answer. (c) Find \(f^{-1}(1)\). (d) Find \(f^{-1}(0)\). (e) Find \(f^{-1}(12)\)

Let \(A_{2}\) be the set of all multiples of 2 except for \(2 .\) Let \(A_{3}\) be the set of all multiples of 3 except for \(3 .\) And so on, so that \(A_{n}\) is the set of all multiples of \(n\) except for \(n,\) for any \(n \geq 2 .\) Describe (in words) the set \(\overline{A_{2} \cup A_{3} \cup A_{4} \cup \cdots}\)

Let \(X=\\{n \in \mathbb{N}: 0 \leq n \leq 999\\}\) be the set of all numbers with three or fewer digits. Define the function \(f: X \rightarrow \mathbb{N}\) by \(f(a b c)=a+b+c\), where \(a, b,\) and \(c\) are the digits of the number in \(X\) (write numbers less than 100 with leading 0 's to make them three digits). For example, \(f(253)=2+5+3=10\) (a) Let \(A=\\{n \in X: 113 \leq n \leq 122\\}\). Find \(f(A)\). (b) Find \(f^{-1}(\\{1,2\\})\) (c) Find \(f^{-1}(3)\). (d) Find \(f^{-1}(28)\). (e) Is \(f\) injective? Explain. (f) Is \(f\) surjective? Explain.

For a given predicate \(P(x)\), you might believe that the statements \(\forall x P(x)\) or \(\exists x P(x)\) are either true or false. How would you decide if you were correct in each case? You have four choices: you could give an example of an element \(n\) in the domain for which \(P(n)\) is true or for which \(P(n)\) if false, or you could argue that no matter what \(n\) is, \(P(n)\) is true or is false. (a) What would you need to do to prove \(\forall x P(x)\) is true? (b) What would you need to do to prove \(\forall x P(x)\) is false? (c) What would you need to do to prove \(\exists x P(x)\) is true? (d) What would you need to do to prove \(\exists x P(x)\) is false?

Draw a Venn diagram to represent each of the following: (a) \(A \cup \bar{B}\) (b) \(\overline{(A \cup B)}\) (c) \(A \cap(B \cup C)\) (d) \((A \cap B) \cup C\) (e) \(\bar{A} \cap B \cap \overline{\mathrm{C}}\) (f) \((A \cup B) \backslash C\)

Suppose \(P(x, y)\) is some binary predicate defined on a very small domain of discourse: just the integers \(1,2,3,\) and \(4 .\) For each of the 16 pairs of these numbers, \(P(x, y)\) is either true or false, according to the following table \((x\) values are rows, \(y\) values are columns). $$\begin{array}{c|cccc} & 1 & 2 & 3 & 4 \\\\\hline 1 & \mathrm{~T} & \mathrm{~F} & \mathrm{~F} & \mathrm{~F} \\ 2 & \mathrm{~F} & \mathrm{~T} & \mathrm{~T} & \mathrm{~F} \\ 3 & \mathrm{~T} & \mathrm{~T} & \mathrm{~T} & \mathrm{~T} \\ 4 & \mathrm{~F} & \mathrm{~F} & \mathrm{~F} & \mathrm{~F}\end{array}$$ For example, \(P(1,3)\) is false, as indicated by the \(\mathrm{F}\) in the first row, third column. Use the table to decide whether the following statements are true or false. (a) \(\forall x \exists y P(x, y)\) (b) \(\forall y \exists x P(x, y)\) (c) \(\exists x \forall y P(x, y)\). (d) \(\exists y \forall x P(x, y)\).