Problem 57 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The truth value of the statement \(∼(p ∧ q) ∨ r\) with \(p\) as false, \(q\) as true, and \(r\) as false is true.

Step 1: Evaluate within the parentheses

Start by evaluating the expression within the parentheses, that is \(p ∧ q\). In this case, \(p\) is false and \(q\) is true. The conjunction operation is true only if both operands are true. Therefore false \(\land\) true results into false.

Step 2: Apply the negation operator

Next, take the result of the conjunction operation and apply the negation operator. The statement \(∼\)false therefore evaluates to true.

Step 3: Evaluate the disjunction operation

Lastly, take the result of the negation operation and apply the disjunction operator with \(r\). This is done by evaluating true \(\vee\) false. The disjunction operation is true if either or both of its operands are true. Therefore, true \(\vee\) false results in true.

Key Concepts

Truth Values

Understanding truth values is crucial when it comes to evaluating logical expressions in mathematics. A statement in logic can only be true or false. These are known as truth values. Every logical statement, regardless of its complexity, will ultimately result in a truth value based on the logical operators involved and the initial truth values of its components.

For instance, if we declare statement p to be false, this is an initial truth value. However, once we combine p with other statements using logical operators, the overall truth value of the resulting expressions can change, depending on the rules governing those operators.

Logical Conjunction

The concept of logical conjunction, typically symbolized as \( \land \) and known simply as 'and', is a fundamental operation in logic. It takes two statements and returns a value of true only if both statements are true. If either statement is false, the conjunction will be false. For example, given statements p and q, \( p \land q \) will only be true when p is true and q is true.

When faced with an expression like \( p \land q \) where p is false and q is true, the conjunction \( false \land true \) is logically false because both conditions are not met.

Logical Disjunction

Opposite to conjunction is the logical disjunction, represented by \( \vee \) and commonly referred to as 'or'. This operator deals with inclusivity. It returns a truth value of true if at least one of its operands is true. In simpler terms, for the expression \( p \vee q \) to be true, either p, q, or both need to be true. The only case where a disjunction is false is when both operands are false.

Using the previous example with p as false and q as true, the disjunction \( false \vee true \) results in a truth value of true. This inclusiveness is what makes disjunction a critical tool in logical reasoning.

Negation Operator

The negation operator, sometimes referred to as 'not', is denoted by \( \sim \) or \( eg \) in logic. It serves to invert the truth value of a statement. If a statement p is true, then \( \sim p \) (not p) will be false; conversely, if p is false, then \( \sim p \) is true. For example, if we apply the negation operator to a false statement, as in our exercise with \( \sim false \) or \( eg false \) we end up with a true value. It effectively flips the truthiness of the given operand.

The negation operator plays a pivotal role in constructing the logical structure of arguments and setting the stage for more complex evaluations, like the one in the given exercise.