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Beginning Activity Beginning Activity 1: Logically Equivalent Statements
A4US

In Exercise 5 and Exercise 6 from Section 2.1, we observed situations where two different statements have the same truth tables. Basically, this means these statements are equivalent, and we make the following definition:

Definition.

Two expressions are logically equivalent provided that they have the same truth value for all possible combinations of truth values for all variables appearing in the two expressions. In this case, we write \(X \equiv Y\) and say that \(X\) and \(Y\) are logically equivalent.

1.

Complete truth tables for \(\mynot \left( {P \wedge Q} \right)\) and \(\mynot P \vee \mynot Q\text{.}\)

2.

Are the expressions \(\mynot \left( {P \wedge Q} \right)\) and \(\mynot P \vee \mynot Q\) logically equivalent?

3.

Suppose that the statement “I will play golf and I will mow the lawn” is false. Then its negation is true. Write the negation of this statement in the form of a disjunction. Does this make sense?

Sometimes we actually use logical reasoning in our everyday living! Perhaps you can imagine a parent making the following two statements.

Statement 1: If you do not clean your room, then you cannot watch TV.
Statement 2: You clean your room or you cannot watch TV.

4.

Let \(P\) be “you do not clean your room,” and let \(Q\) be “you cannot watch TV.” Use these to translate Statement 1 and Statement 2 into symbolic forms.

5.

Construct a truth table for each of the expressions you determined in Exercise 4. Are the expressions logically equivalent?

6.

Assume that Statement 1 and Statement 2 are false. In this case, what is the truth value of \(P\) and what is the truth value of \(Q\text{?}\) Now, write a true statement in symbolic form that is a conjunction and involves \(P\) and \(Q\text{.}\)

7.

Write a truth table for the (conjunction) statement in Exercise 6 and compare it to a truth table for \(\mynot \left( {P \to Q} \right)\text{.}\) What do you observe?