91Ó°ÊÓ

Let \(t\) be a tautology and \(p\) an arbitrary proposition. Find the truth value of each. $$(\sim t) \rightarrow p$$

Let \(t\) be a tautology and \(p\) an arbitrary proposition. Find the truth value of each. $$p \rightarrow t$$

Use De Morgan's laws to verify each. (Hint: \(p \rightarrow q \equiv \sim p \vee q\) ).

Let UD \(=\) set of integers, \(\mathrm{P}(x, y) : x\) is a multiple of \(y,\) and \(Q(x, y) : x \geq y\) Determine the truth value of each proposition. $$(\forall x)(\exists y) P(x, y)$$

Let UD = set of integers, \(P(x, y): x\) is a multiple of \(y,\) and \(Q(x, y): x \geq y\) Determine the truth value of each proposition. $$(\forall x)(\exists y) Q(x, y)$$

Let UD = set of integers, \(P(x, y): x\) is a multiple of \(y,\) and \(Q(x, y): x \geq y\) Determine the truth value of each proposition. $$(\forall x)[\mathrm{P}(x, 3) \rightarrow \mathrm{Q}(x, 3)]$$

Let UD \(=\) set of integers, \(\mathrm{P}(x, y) : x\) is a multiple of \(y,\) and \(Q(x, y) : x \geq y\) Determine the truth value of each proposition. $$(\forall x) | P(x, 3) \rightarrow Q(x, 3) ]$$

Show that the connectives \(\wedge, \rightarrow,\) and \(\leftrightarrow\) can be expressed in terms of v and \(\sim .\) (Hint: Use Exercise 44, law 18, and Tables 1.6 and 1.7.)

Let \(t\) be a tautology and \(p\) an arbitrary proposition. Find the truth value of each. $$(p \vee t) \rightarrow t$$

Simplify each boolean expression. $$p \wedge(p \wedge q)$$