Questions: Use a truth table to determine whether the statement is a tautology, a self-contradiction, or neither [ (sim p vee q) wedge(p wedge sim q) ] Complete the truth table. p q sim p vee q p wedge sim q (sim p vee q) wedge(p wedge sim q) T T

Solution Steps

To determine whether the given statement is a tautology, a self-contradiction, or neither, we need to complete the truth table for the expression \((\sim p \vee q) \wedge (p \wedge \sim q)\). We will evaluate each component of the expression for all possible truth values of \(p\) and \(q\), and then determine the truth value of the entire expression for each case. A tautology will have all true values, a self-contradiction will have all false values, and neither will have a mix of true and false values.

To determine the nature of the statement \((\sim p \vee q) \wedge (p \wedge \sim q)\), we first evaluate each component for all possible truth values of \(p\) and \(q\). The components are:

• \(\sim p \vee q\)
• \(p \wedge \sim q\)

We construct the truth table by evaluating the components and the entire expression for each combination of \(p\) and \(q\):

• For \(p = \text{True}, q = \text{True}\):
  • \(\sim p \vee q = \text{True}\)
  • \(p \wedge \sim q = \text{False}\)
  • \((\sim p \vee q) \wedge (p \wedge \sim q) = \text{False}\)
• For \(p = \text{True}, q = \text{False}\):
  • \(\sim p \vee q = \text{False}\)
  • \(p \wedge \sim q = \text{True}\)
  • \((\sim p \vee q) \wedge (p \wedge \sim q) = \text{False}\)
• For \(p = \text{False}, q = \text{True}\):
  • \(\sim p \vee q = \text{True}\)
  • \(p \wedge \sim q = \text{False}\)
  • \((\sim p \vee q) \wedge (p \wedge \sim q) = \text{False}\)
• For \(p = \text{False}, q = \text{False}\):
  • \(\sim p \vee q = \text{True}\)
  • \(p \wedge \sim q = \text{False}\)
  • \((\sim p \vee q) \wedge (p \wedge \sim q) = \text{False}\)

From the truth table, we observe that the expression \((\sim p \vee q) \wedge (p \wedge \sim q)\) is false for all combinations of \(p\) and \(q\). This indicates that the statement is a self-contradiction.

Final Answer