From now onwards, we are going to start with the very first chapter of discrete mathematics that is propositional logic. In this lecture, we are going to talk about the motivation and introduction to propositional logic. The phrase propositional logic is composed of two words.
We first have to note down this point that propositional logic is composed of two words. One is preposition and the other one is logic. Simple.
Now, let's try to understand what is logic? Logic is the science of reasoning. It helps us to understand and reason about different mathematical statements. This is very important.
We would be able to reason about different mathematical statements with the help of logic. With the rules of logic, we would be able to think about mathematical statements and finally we would be able to prove or disprove those mathematical statements precisely. That is what we want finally, right? First of all, we try to think about the mathematical statements with the help of logic.
And finally we would be able to prove or disprove those mathematical statements precisely. That is what we want finally, right? Let's consider one example. Rules of logic enable us to reason about the mathematical statements like for every positive integer n, the sum of positive integers not exceeding n is n into n plus one divided by two. We are well familiar with this particular formula which is nothing but the sum of first n natural numbers or sum of first n positive integers.
Right? We know that this particular mathematical statement is a valid mathematical statement. And we would be able to prove this particular statement with rules of logic.
Either it is a valid mathematical argument or it is an invalid mathematical argument. But we know that this mathematical argument is a valid mathematical argument. And we can prove this also with the help of logic.
This is what a logic is all about. With logic, we would be able to reason about statements, we would be able to think about mathematical statements and finally we would be able to prove them or disprove them by using certain rules that are provided by logic itself. Now let's try to understand what is the purpose of logic. Purpose of logic is to construct valid arguments. First of all, this is very important point to note.
This is already we know that Purpose of logic is to construct valid arguments which are also called as proofs. Once we prove a mathematical statement is true, then we call it a theorem. We already know that.
When we prove a mathematical statement is true, then we call it a theorem. And this is the basis of whole mathematics. Right? This is the basis of whole mathematics.
Therefore, logic is very very important. Let's consider one more example. Here in this example, we are talking about one famous puzzle called Knights and Knaves puzzle.
In an island, there are two kinds of inhabitants. Knights who always tell the truth and their opposites, Knaves, who always lie. You encounter two people, Alice and Bob. Who are Alice and Bob? If Alice says Bob is a knight and Bob says the two of us are opposite types.
We want to know that who are Alice and Bob? If Alice says that Bob is knight and Bob says two of us are opposite types. These kinds of puzzles we would be able to solve very easily using an area of logic called prepositional logic. Please note down. With prepositional logic we would be able to solve these types of puzzles very easily and in a systematic manner.
that is the application of propositional logic. Therefore, learning propositional logic is fun and is very important. Now, let's try to understand what is propositional logic. We already know that propositional logic is an area of logic.
But in order to understand what is propositional logic, we need to understand what is proposition. Proposition is a declarative sentence. Please note down. Proposition is a declarative sentence. Declarative sentence is a sentence that is declaring a fact or stating an argument which can be either true or false but cannot be both.
Please note down this point. Proposition is a declarative sentence which can be either true or false. But it cannot be both. For example, Delhi is the capital of India.
It can be true or it can be false. Water froze this morning. Definitely this statement can be true or false. One plus one equals to two.
We know that this statement is true. Also, it is a declarative sentence. Therefore, this is a preposition.
Now, let's consider some more sentences which are not prepositions. What time is it? This is not a preposition because this is not a declarative sentence. It is asking something.
What time is it? It is not declaring a fact. Therefore, it is not a preposition.
x plus one equals two. It might seems like it is a declarative sentence. But think about it. It can be both true or false.
Maybe you provide x equals to one. In that case, this statement is true. But if you provide x equals to two, then this statement is false.
Therefore, we are not certain about this statement. It totally depends on the value of x. Therefore, we cannot say that it is a preposition. As we know, Preposition is a declarative sentence that can be true or false. But it cannot be both.
Send us your resume before 11 pm. You are commanding someone to send the resume before 11 pm. It is not a declarative sentence and hence it is not a preposition. And also it doesn't make sense that this particular statement is either true or false.
Right? I request you to please allow me a day off. This particular statement is definitely not a declarative sentence because you are requesting something.
You are not declaring some kind of fact or stating an argument. Hence, this is not a preposition. Fetch my umbrella. Obviously, you are commanding someone to fetch your umbrella.
How it can be a declarative sentence then? Definitely, it is not a preposition. Ok, now let's consider one homework problem.
Which of the following is our prepositions? Also, if you find the preposition, state whether it is true or false. London is in Denmark. Do your homework.
India wins the match by two runs. X is an even number. Five is an odd number. Rahul, five plus seven equals ten.
The moon is made of cheese. The only odd prime number is two. God bless you.
You can always post your answers in the comment section below. Okay friends, this is it for now. Thank you for watching this lecture.