Which of these sentences are propositions? What are the truth values of those that are propositions?

/ Propositional Logic / By

Propositional logic is a very important topic in Discrete Mathematics. It is part of the foundations every student should know. In this post, I’ll show how I solve this specific type of exercise.

In mathematics, definitions are very important. As I always recommend to my students, when you are starting a new topic, and you are solving exercises, make sure you have all the definitions you need at hand.

In this case, we should answer which of the sentences are propositions and the truth values of each one. Therefore, we only need the definition of propositions and truth value.

Table of Contents

Definitions

A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.

The truth value of a proposition is whether the statement is true or false.

Some beginners get confused with this. They think that because the statement is false it is not a proposition. This is not the case, as per the definition a proposition can be false.

All the definitions I use are from the book Discrete Mathematics and its Applications by Rosen.

Now, let’s answer the following exercise.

Which of these sentences are propositions? What are the truth values of those that are propositions?


a) Boston is the capital of Massachusetts

Is this a declarative sentence? Yes.

Can it be true or false, but not both? Yes.

Then, we can affirm that the sentence above is a proposition.

For the second part of the questions, we should answer what the truth value is. We know that Boston is the capital of Massachusetts. Therefore, the truth value of the proposition is True.

Answer: Yes, it is a proposition, and it is true.

b) Miami is the capital of Florida

As in the case before, we make use of the definition as follows.

Is this a declarative sentence? Yes.

Can it be true or false, but not both? Yes.

Then, we can affirm that the sentence above is a proposition.

In this case, we know that the capital of Florida is Tallahassee. Because of this, we can state that the truth value is false.

Answer: Yes, it is a proposition, and it is false.

c) 2+3=5

Is this a declarative sentence? Yes.

Can it be true or false, but not both? Yes.

Then, we can affirm that the sentence above is a proposition.

We also know from mathematics that 2 plus 3 equals 5. So, the truth value is True.

Answer: Yes, it is a proposition, and it is true.

d) 5+7=10


Is this a declarative sentence? Yes.

Can it be true or false, but not both? Yes.

Then, we can affirm that the sentence above is a proposition.

We also know from mathematics that 5 plus 7 equals 12. So, the truth value is False.

Answer: Yes, it is a proposition, and it is false.

e) x+2=11


Is this a declarative sentence? Yes.

Can it be true or false, but not both? No.

The truth value of this sentence will depend on the value that we assign to the variable x. So, the truth value can be both, depending on the value of x.

Then, we can affirm that the sentence above is not a proposition.

Answer: It is not a proposition.

f) Answer this question

Is this a declarative sentence? Yes.

Can it be true or false, but not both? No.

The result of the declarative sentence above is neither true nor false. Because of that, we can affirm that the sentence above is not a proposition.

Answer: It is not a proposition.

Summary

Always remember to have all the definitions you need at hand (or memorized) when solving mathematics problems.

In the case of the type of exercise where you are asked whether a sentence is a proposition or not, always follow the definition.

First, answer if is a declarative sentence. Then, answer if it can be true or false but not both.

Lastly, if it is a proposition, use the knowledge you already have about the declarative sentence to determine what is the truth value.

Related posts: