91Ó°ÊÓ

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. $$ \neg(\forall x P(x)) $$

Write the truth table of each proposition. $$ (p \wedge q) \vee(\neg p \vee q) $$

Tell which rule of inference is used. Fishing is a popular sport. Therefore, fishing is a popular sport or lacrosse is wildly popular in California.

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 \(\varnothing\). $$\bar{X} \cup \bar{Y}$$

What is the cardinality of \(\varnothing ?\)

Refer to the propositions \(p, q,\) and \(r ; p\) is true, \(q\) is false, and r's status is unknown at this time. Tell whether each proposition is true, is false, or has unknown status at this time. $$ r \rightarrow q $$

Write the truth table of each proposition. $$ \neg(p \wedge q) \vee(r \wedge \neg p) $$

Tell which rule of inference is used. If fishing is a popular sport, then lacrosse is wildly popular in California. Fishing is a popular sport. Therefore, lacrosse is wildly popular in California.

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. $$ \exists x P(x) $$

Refer to the propositions \(p, q,\) and \(r ; p\) is true, \(q\) is false, and r's status is unknown at this time. Tell whether each proposition is true, is false, or has unknown status at this time. $$ (p \wedge r) \leftrightarrow r $$