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Chapter 3: Problem 21

Construct a truth table for the given statement. \((p \leftrightarrow q) \rightarrow p\)

Short Answer

Expert verified
The truth table for the statement is as follows: For TT we get T, for TF we get T, for FT we get T, and finally for FF we get F. This means that the entire logical expression is true for all cases except when both p and q are false.

Step by step solution

01

Preparation of the truth values for p and q

First, write down all possible combinations of truth values for p and q. There are four possibilities, because each of p and q can be either true (T) or false (F). The combinations are: TT, TF, FT, FF.
02

Calculation of the biconditional (iff) operation

Next, calculate the biconditional operation \((p \leftrightarrow q)\) for each combination of p and q. For TT it is T, for TF it is F, for FT it is F, and for FF it is T.
03

Calculation of the resultant implication operation

After calculating the biconditional for each pair (p, q), calculate the result of the complete logical expression, which is the implication \((p \leftrightarrow q) \rightarrow p\). Whenever the first part \(p \leftrightarrow q\) is true, then p should also be true for the entire expression to be true. Whenever the first part is false, it doesn't matter whether p is true or false, the result will always be true (this is how logical implication works).

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Most popular questions from this chapter

Determine whether each argument is valid or invalid. No \(A\) are \(B\), no \(B\) are \(C\), and no \(C\) are \(D\). Thus, no \(A\) are \(D\).

Use the standard forms of valid arguments to draw a valid conclusion from the given premises. If all houses meet the hurricane code, then none of them are destroyed by a category 4 hurricane. Some houses were destroyed by Andrew, a category 4 hurricane. Therefore, ...

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If I'm tired, I'm edgy. If I'm edgy, I'm nasty. \(\therefore\) If I'm tired, I'm nasty.

Write an original argument in words that has a true conclusion, yet is invalid.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) There must be a dam or there is flooding. This year there is flooding. \(\therefore\) This year there is no dam.
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