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Chapter 3: Problem 61

Determine the truth value for each statement when \(p\) is false, \(q\) is true, and \(r\) is false. \((\sim p \wedge q) \vee(\sim r \wedge p)\)

Short Answer

Expert verified
The truth value of the statement is true.

Step by step solution

01

Negation of variables

The operation to be performed first will be negation since it takes precedence, so replace \(\sim p\) and \(\sim r\) with their values. The negation of \(p\) (which is false) will be true and the negation of \(r\) (which is false) will be true.
02

Evaluation of conjunctions

Next, evaluate the value of each conjunction. The conjunction operation (\(\wedge\)) returns true if both of its operands are true. So, \((\sim p \wedge q)\) returns true (since \(\sim p\) and \(q\) are both true) and \((\sim r \wedge p)\) returns false (since \(\sim r\) is true but \(p\) is false).
03

Evaluation of disjunction

Finally, evaluate the disjunction (\(\vee\)) operation. The disjunction operation returns true if either of its operands is true. So, \((\sim p \wedge q) \vee(\sim r \wedge p)\) returns true, since the first operand is true.

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