91Ó°ÊÓ

Which of these sentences are propositions? What are the truth values of those that are propositions? a) Boston is the capital of Massachusetts. b) Miami is the capital of Florida. c) \(2+3=5\) d) \(5+7=10\) e) \(x+2=11\) f) Answer this question.

Prove that \(n^{2}+1 \geq 2^{n}\) when \(n\) is a positive integer with \(1 \leq n \leq 4 .\)

Find the argument form for the following argument and determine whether it is valid. Can we conclude that the conclusion is true if the premises are true? If Socrates is human, then Socrates is mortal. Socrates is human. ? Socrates is mortal.

Which of these are propositions? What are the truth values of those that are propositions? a) Do not pass go. b) What time is it? c) There are no black flies in Maine. d) \(4+x=5\) . e) The moon is made of green cheese. f) \(2^{n} \geq 100\) .

Find the argument form for the following argument and determine whether it is valid. Can we conclude that the conclusion is true if the premises are true? If George does not have eight legs, then he is not a spider. George is a spider. ? George has eight legs.

Translate these statements into English, where the domain for each variable consists of all real numbers a) \(\exists x \forall y(x y=y)\) b) \(\forall x \forall y(((x \geq 0) \wedge(y<0)) \rightarrow(x-y>0))\) c) \(\forall x \forall y \exists z(x=y+z)\).

Let \(P(x)\) be the statement "The word \(x\) contains the letter \(a\) ." What are these truth values? \(\begin{array}{ll}{\text { a) } P(\text { orange })} & {\text { b) } P(\text { lemon })} \\ {\text { c) } P(\text { true })} & {\text { d) } P(\text { false })}\end{array}\)

Use a direct proof to show that the sum of two even integers is even.

Use a proof by cases to show that 10 is not the square of a positive integer. [Hint: Consider two cases: \((i) 1 \leq x \leq 3\) , (ii) \(x \geq 4.1\)

Show that \(\neg(\neg p)\) and \(p\) are logically equivalent.