91Ó°ÊÓ

Tell whether the statement is a propositional function. For each statement that is a propositional function, give a domain of discourse. \((2 n+1)^{2}\) is an odd integer.

Let the universe be the set \(U=\\{1,2,3, \ldots, 10\\}\) Let \(A=\\{1,4,7,10\\}, B=\\{1,2,3,4,5\\},\) and \(C=\\{2,4,6,8\\} .\) List the elements of each set. $$A \cup B$$

Formulate the arguments of Exercises \(6-9\) symbolically and deter. mine whether each is valid. P: The Democrats win. \(q:\) The Republicans win. \(r:\) Unemployment is up. s: The economy is up If the Democrats win, then unemployment is up or the economy is up. If the Republicans win, then unemployment is up.The economy is not up. The Democrats win \- Unemployment is up or the Republicans win.

In Exercises \(10-14\), write the given argument in words and deter. mine whether each argument is valid. Let p: 4 gigabytes is better than no memory at all. q: We will bury more memory. \(r:\) We will bury a new computer. $$ \begin{array}{l} P \rightarrow(r \vee q) \\ \frac{r \rightarrow \neg q}{\therefore p \rightarrow r} \end{array} $$

Refer to a coin that is flipped 10 times. Write the negation of the proposition. Some heads and some tails were obtained.

Let the universe be the set \(Z^{+} .\) Let \(X=\) \\{1,2,3,4,5\\} and let \(Y\) be the set of positive, even integers. In set builder notation, \(Y=\left\\{2 n \mid n \in Z^{+}\right\\} .\) In Exercises \(18-27,\) give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as \(\varnothing\). Describe \(\bar{Y}\) in words.

Let the universe be the set \(Z^{+} .\) Let \(X=\) \\{1,2,3,4,5\\} and let \(Y\) be the set of positive, even integers. In set builder notation, \(Y=\left\\{2 n \mid n \in Z^{+}\right\\} .\) In Exercises \(18-27,\) give a mathematical notation for the set by listing the elements if the set is finite, by using set-builder notation if the set is infinite, or by using a predefined set such as \(\varnothing\). $$X \cap Y$$

Suppose that the domain of discourse of the propositional function \(P\) is \(\\{1,2,3,4\\} .\) Rewrite each propositional function using only negation, disjunction, and conjunction. $$ \forall x P(x) $$

Give an argument using rules of inference to show that the conclusion follows from the hypotheses. Hypotheses: If there is gas in the car, then I will go to the store. If I go to the store, then I will get a soda. There is gas in the car. Conclusion: I will get a soda.

Determine the truth value of each proposition. If \(3+5<2,\) then \(1+3 \neq 4\).