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Chapter 2: Problem 6

Construct a truth table for each of the following statements: (a) \(P \vee \neg Q\) (c) \(\neg P \vee \neg Q\) (b) \(\neg(P \vee Q)\) (d) \(\neg P \wedge \neg Q\) Do any of these statements have the same truth table?

Short Answer

Expert verified
The truth tables for the given statements are as follows: (a) \(P \vee \neg Q\): P | Q | 卢Q | P 鈭� 卢Q T | T | F | T T | F | T | T F | T | F | F F | F | T | T (c) \(\neg P \vee \neg Q\): P | Q | 卢P | 卢Q | 卢P 鈭� 卢Q T | T | F | F | F T | F | F | T | T F | T | T | F | T F | F | T | T | T (b) \(\neg(P \vee Q)\): P | Q | P 鈭� Q | 卢(P 鈭� Q) T | T | T | F T | F | T | F F | T | T | F F | F | F | T (d) \(\neg P \wedge \neg Q\): P | Q | 卢P | 卢Q | 卢P 鈭� 卢Q T | T | F | F | F T | F | F | T | F F | T | T | F | F F | F | T | T | T No two statements have the same truth table, hence none are logically equivalent.

Step by step solution

01

List all possible values of P and Q

饾憙 and Q are logical statements that can either be true (T) or false (F). We need to consider all possible combinations of T and F values, which are (T, T), (T, F), (F, T), and (F, F).
02

Construct the truth tables

We will create truth tables for each of the given logical statements using the combinations of P and Q from Step 1. For (a) \(饾憙 \vee \neg Q\): P | Q | 卢Q | P 鈭� 卢Q T | T | F | T T | F | T | T F | T | F | F F | F | T | T For (c) \(\neg P \vee \neg Q\): P | Q | 卢P | 卢Q | 卢P 鈭� 卢Q T | T | F | F | F T | F | F | T | T F | T | T | F | T F | F | T | T | T For (b) \(\neg(饾憙 \vee 饾憚)\): P | Q | P 鈭� Q | 卢(P 鈭� Q) T | T | T | F T | F | T | F F | T | T | F F | F | F | T For (d) \(\neg 饾憙 \wedge \neg 饾憚\): P | Q | 卢P | 卢Q | 卢P 鈭� 卢Q T | T | F | F | F T | F | F | T | F F | T | T | F | F F | F | T | T | T
03

Compare the truth tables

Now, let's compare the truth tables. If any of the statements have the same truth table, it means they are logically equivalent. As we can see from the truth tables, none of the given statements have the same truth table. Therefore, no two statements are logically equivalent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logical Equivalence
Logical equivalence is a fundamental concept in propositional logic where two statements are considered equivalent if they always yield the same truth value in every possible scenario. In simpler terms, no matter what the actual truths of the individual pieces are, if the overall statements give the same result, they're logically equivalent.

Think of it like two different math problems that always result in the same answer, regardless of the numbers you start with. To determine logical equivalence, we use truth tables to compare statements directly. If the columns representing the overall truth of two statements match exactly, they are logically equivalent. In the exercise provided, none of the statements shared a truth table, which means none of them were logically equivalent to each other.
Negation Logic
Negation logic is all about flipping the truth value of a statement. If a statement is true, its negation is false, and vice versa. This is like saying 'It's not raining' when indeed it is sunny outside. In symbolism, this is represented as 'eg'.

In our exercise, negation plays a crucial role in changing the truth values of statements P and Q. When constructing the truth table for eg P or eg Q, we first determine the truth of P and Q, then simply flip it to get their negations. You can see this in action in parts (b) and (d) of the exercise, showing the power of negation in altering the outcome.
Disjunction Logic
Disjunction logic refers to a scenario where at least one of the statements needs to be true for the overall expression to be true. Symbolised as 'vee', this is often referred to in layman's terms as 'or'. It embodies the idea of inclusivity in options.

To apply disjunction logic to a truth table, as seen in parts (a) and (c) of the exercise, you check if at least one input column is true. If so, then the statement evaluates to true. Remember, it only takes one 'true' to make the whole disjunction true. That's why in a rain-or-shine event, the event happens regardless of whether it's raining, because shine (or no rain) is still on the table.
Conjunction Logic
Conjunction logic is the strict counterpart to disjunction. It requires both statements to be true for the whole to be true. Symbolically, we use 'wedge', and it's comparable to the word 'and' in everyday language. If you have a condition like 'You need a ticket AND an ID to enter', both requirements must be met.

When filling out the truth table for a conjunction, as done in part (d) of the exercise, you look for the case where both contributing statements are true. If even one of them is false, the conjunction is false. It's a bit like having two keys for one lock - missing one means you're not getting in.

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Most popular questions from this chapter

Suppose that \(P\) and \(Q\) are statements for which \(P \rightarrow Q\) is true and for which \(\neg Q\) is true. What conclusion (if any) can be made about the truth value of each of the following statements? (a) \(P\) (b) \(P \wedge Q\) (c) \(P \vee Q\)

Complete the following table: $$ \begin{array}{|l|c|c|c|} \hline \text { English Form } & \text { Hypothesis } & \text { Conclusion } & \text { Symbolic Form } \\ \hline \text { If } P, \text { then } Q . & P & Q & P \rightarrow Q \\ \hline Q \text { only if } P . & Q & P & Q \rightarrow P \\ \hline P \text { is necessary for } Q . & & & \\ \hline P \text { is sufficient for } Q . & & & \\ \hline Q \text { is necessary for } P . & & & \\ \hline P \text { implies } Q . & & & \\ \hline P \text { only if } Q . & & & \\ \hline P \text { if } Q . & & & \\ \hline \text { If } Q \text { then } P . & & & \\ \hline \text { If } \neg Q, \text { then } \neg P . & & & \\ \hline \text { If } P, \text { then } Q \wedge R . & & & \\ \hline \text { If } P \vee Q, \text { then } R . & & & \\ \hline \end{array} $$

Suppose that \(P\) and \(Q\) are statements for which \(P \rightarrow Q\) is false. What conclusion (if any) can be made about the truth value of each of the following statements? (a) \(\neg P \rightarrow Q\) (b) \(Q \rightarrow P\) (c) \(P \vee Q\)

Let \(a\) be a real number and let \(f\) be a real-valued function defined on an interval containing \(x=a\). Consider the following conditional statement: If \(f\) is differentiable at \(x=a,\) then \(f\) is continuous at \(x=a\). Which of the following statements have the same meaning as this conditional statement and which ones are negations of this conditional statement? (a) If \(f\) is continuous at \(x=a,\) then \(f\) is differentiable at \(x=a\). (b) If \(f\) is not differentiable at \(x=a,\) then \(f\) is not continuous at \(x=a\). (c) If \(f\) is not continuous at \(x=a,\) then \(f\) is not differentiable at \(x=a\). (d) \(f\) is not differentiable at \(x=a\) or \(f\) is continuous at \(x=a\). (e) \(f\) is not continuous at \(x=a\) or \(f\) is differentiable at \(x=a\). (f) \(f\) is differentiable at \(x=a\) and \(f\) is not continuous at \(x=a\).

Use truth tables to prove the following logical equivalency from Theorem 2.8: $$ [(P \vee Q) \rightarrow R] \equiv(P \rightarrow R) \wedge(Q \rightarrow R) $$
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