Propositional Logic in Discrete mathematics

Propositional logic can be described as a simple form of logic where propositions are used to create all the statements. The proposition can be described as a declarative statement, which means it is used to declare some facts. The statements of propositional logic can either be true or false, but they cannot be both.

Examples of Propositional Logic

There are various examples of propositional logic, and some of them are described as follows:

• 5 + 2 = 7
• Bananas are green.
• Yogi Adityanath is the chief minister of Utter Pradesh
• Five and five make eight
• 2021 is the worst year
• Delhi is the capital of US.
• Karnataka is in India.

Here,

All the above statements are either true or false, but they can't be both. That's why these statements are propositions.

Types of Propositions

Propositional logic is used to contain two types of propositions, which are described as follows:

1. Atomic Propositions
2. Compound Propositions

Atomic Propositions

The propositions will be known as the atomic propositions if it will not further divided. This type of proposition is also known as the simple proposition. It is used to have a single proposition symbol. Atomic propositions are indicated by small letters such as p, q, r, s, etc. The sentences of atomic propositions can either be true or false.

Examples of Atomic propositions

Some examples of atomic propositions are described as follows:

• p: 5 + 3 = 8. It is an atomic proposition because it is a true fact.
• q: Bananas are yellow. It is an atomic proposition because it is a true fact.
• r: Sun rises in the west. It is an atomic proposition because it is a false fact.
• s: Moon is black. It is an atomic proposition because it is a true fact.

Compound Propositions

The propositions will be known as compound propositions if they are formed by the combination of one or more atomic propositions with the help of connectives. In other words, we can say that if a proposition has some connectives, then it will be known as a compound proposition. We can construct this proposition by the combination of simple and atomic propositions with the help of parenthesis and logical connectives. Compound propositions are indicated with the help of capital letters such as P, Q, R, S, etc.

Examples of Compound Propositions

Some examples of compound propositions are described as follows:

• It is sunny today and I will go to school.
• Oranges are orange and bananas are yellow.
• Sun sets in the west and sun rises in the east.
• John is an engineer and he works for the XYZ Company.

Statements that are not Propositions

There are some statements that are not propositions, which are shown below:

Command: A statement will hold command if one person is being told to do something. In this type of statement, someone tells the other one to do something, so these statements may or may not begin with an imperative (bossy) verb.

Questions: A statement can be described as a sentence that is used to tell us something. A question can be described as a sentence that asks us something. In other words, a statement will hold a question if one person asks something from another person. In this type of statement, a question needs an answer.

Exclamation: We can also call the exclamation sentence as an exclamative clause or exclamatory sentence, which can be described as a statement that shows strong emotion. In other words, a statement will hold an exclamation if it shows strong emotions. Typically an exclamation mark is used at the end of an exclamation sentence.

Inconsistent: A statement will be inconsistent if we criticize someone for not behaving in the same way, every time a similar situation occurs. For example: I always tell tie.

Predicate: A predicate can be described as a sentence that is used to contain a finite number of variables. When we assign a specific value to the variable, this sentence becomes a statement.

For examples:

In this example, we will show some statements which are not propositions. These statements are described as follows:

1. Catch the ball. This statement is a command because here, one person is telling another one to go to catch the ball.
2. Do you have any problem with Harry. This statement is a question because here, one person asks a question.
3. What an awesome day!. This statement is an exclamation because there is an emotion in this statement, and this statement also has an exclamation symbol !.
4. I always tell lies. This statement is inconsistent because one person can never tell a lie every time.
5. P(x) = a - 5 = 10. This statement is a predicate because it contains a variable x, and the value is assigned to that variable.

Examples of Propositions

Here we will explain various examples of propositions, and some of them are described as follows:

1. Lucknow is the capital of Uttar Pradesh. This is a true proposition.
2. 2024 will be the leap year. This is a true proposition.
3. Mangoes are black. This is a false proposition.
4. P(x) = x + 2 = 4. This is not a proposition because it is a predicate.
5. P(5): 4 + 8 = 10. This is a false proposition because 4+8 does not equal to 10.
6. Bananas are green. This is a false proposition.
7. Oranges are purple. This is a false proposition.
8. Sum of three and five is eight. This is a true proposition.
9. X is less than 4. This is not a proposition. This is a predicate because there is a variable x that does not assign to any value.
10. Go to college. This is not a proposition. This is a command because one person is telling someone to do the thing.
11. Are you sure? This is not a proposition. This is a question because here, one person asks a question.
12. Wow! She is very beautiful. This is not a proposition. This is an exclamation because there is a strong emotion, and it contains an exclamation symbol.
13. Delhi is not part of India. This is a false proposition.
14. I am a good student. This is a proposition that can either be true or false.
15. I always tell a lie. This is not a proposition. This is inconsistent because one person cannot tell a lie every time.
16. This statement is true. This is a true proposition.
17. This statement is false. This is not a proposition. This is an inconsistent statement.
18. Water is running. This is a proposition that can either be true or false.
19. Sun will rise tomorrow. This is a proposition that can either be true or false.
20. Don't touch my cheeks. This is not a proposition. This is a command because one person tells someone to do a thing.
21. Don't go there. This is not a proposition. This is a command because one person tells someone to do a thing.

Limitations of Propositional Logic

There are various limitations of propositional logic, and some of them are shown below:

1. In propositional logic, we cannot show the relations such as some, ALL, or none. For example:
   • All girls in my class are smart and intelligent.
   • Some mangoes are sweet and tart.
2. There is limited expressive power in propositional logic.
3. The statements of propositional logic cannot be indicated in the form of their properties or logical relationships.

Important Points of Propositional logic

There are some important points that are related to the propositional logic, which are described as follows:

• We can also call propositional logic as Boolean logic. This is because the propositional logic also works on 1 and 0.
• In propositional logic, we can indicate logic with the help of symbolic variables, and we can indicate the propositions with the help of any symbol like P, Q, R, X, Y, Z, etc
• Propositional logic can be indicated as either true or false, but we cannot indicate it in both ways.
• It is used to have relations or functions, objects, and logical connectives, which are also known as logical operators.
• The basic elements of propositional logic are propositions and connectives.
• If there is a propositional formula, which is always true, then it will be known as a tautology. This type of formula is also known as a valid sentence.
• If there is a propositional formula, which is always false, then it will be known as a contradiction.
• If there is a propositional formula, which is true and false, then it will be known as consistent or Contingency.