let p represent a true statement, while q and r represent false statements, find the truth value of the
compound statement ~[(p^q)^r]
p^q = T^F = F
F^r = F no matter what r is
~F = T
The equation (x + 6)2 + (y + 4)2 = 36 models the position and range of the source of a radio signal. Describe the position of the source and the range of the signals.
To find the truth value of the compound statement ~[(p^q)^r], we need to break it down into steps:
1. Let's start by evaluating the innermost expression (p^q), which represents the conjunction (AND) of p and q.
Since q is a false statement, the conjunction (p^q) will be false regardless of the truth value of p.
Therefore, (p^q) is false.
2. Next, let's evaluate the expression [(p^q)^r], which represents the conjunction (AND) of (p^q) and r.
Since (p^q) is false and r is also false, the conjunction [(p^q)^r] will be false.
3. Finally, we need to negate the entire expression using the tilde symbol (~) to get the final truth value.
Since [(p^q)^r] is false, its negation ~[(p^q)^r] will be true.
So, the truth value of the compound statement ~[(p^q)^r] is true.
To find the truth value of the compound statement ~[(p^q)^r], we need to understand the truth values for each component of the statement.
Let's break it down step by step:
1. ~[(p^q)^r]: The tilde (~) represents the negation or the logical NOT operator. It flips the truth value of a statement.
2. (p^q): The caret (^) represents the logical AND operator. It returns true if both statements on either side are true.
3. [(p^q)^r]: We combine the logical AND operator (^) with statement r using parentheses to give [(p^q)^r].
Now, let's determine the truth value of each statement involved:
- p represents a true statement.
- q represents a false statement.
- r represents a false statement.
Next, let's evaluate the innermost expression, (p^q):
Since q is false, (p^q) will be false regardless of the value of p. Thus, (p^q) is false.
Now, substituting the values into [(p^q)^r]:
[(p^q)^r] = [false^r] = false
Finally, applying the negation (~) to false:
~[false] = true
Therefore, the truth value of the compound statement ~[(p^q)^r] is true.