Implication in Discrete mathematics

An implication statement can be represented in the form "if....then". The symbol ⇒ is used to show the implication. Suppose there are two statements, P and Q. In this case, the statement "if P then Q" can also be written as P ⇒ Q or P → Q, and it will be read as "P implies Q". In this implication, the statement P is a hypothesis, which is also known as premise and antecedent, and the statement Q is conclusion, which is also known as the consequent.

The implication also plays an important role in the logical argument. If the implication of the statements is known to be true, then whenever the premise is met, the conclusion must also be true. Because of this reason, the implication is also known as the conditional statement.

Some examples of implications are described as follows:

• "If the weather of GOA is sunny, then we will go to beach".
• "If the club has discount system, then we will go to that club".
• "If it is sunny while going beach, then we will be tanned".

The logical implication can be expressed in various ways, which are described as follows:

1. If p then q
2. If p, q
3. q when p
4. Q only if P
5. q unless ~p
6. q whenever p
7. p is a sufficient condition for q
8. q follow p
9. p implies q
10. A necessary condition for p is q
11. q if p
12. q is necessary for p
13. p is a necessary condition for q

Now we will describe the examples of all the above-described implications with the help of premise P and conclusion Q. For this, we will assume that P = It is sunny and Q = I will go to the beach.

P ⇒ Q

1. IF it is sunny THEN I will go to the beach
2. IF it is sunny, I will go to the beach
3. I will go to the beach WHEN it is sunny
4. I will go to the beach ONLY IF it is sunny
5. I will go to the beach UNLESS it is not sunny
6. I will go to the beach WHENEVER it is sunny
7. It is sunny IS A SUFFICIENT CONDITION FOR I will go to the beach
8. I will go to the beach FOLLOW it is sunny
9. It is sunny IMPLIES I will go to the beach
10. A NECESSARY CONDITION FOR it is sunny is I will go to the beach
11. I will go to the beach IF it is sunny
12. I will go to the beach IS NECESSARY FOR it is sunny
13. It is sunny IS A NECESSARY CONDITION FOR I will go to the beach

When there is a conditional statement "if p then q", then this statement P ⇒ Q will be false when Premises p is true, and Conclusion q is false. In all the other cases, that means when p is false or Q is true, the statement P ⇒ Q will be true. We can represent this statement with the help of a truth table in which the false will be represented by F and true will be represented by T. The truth table of the statement "if P then Q" is described as follows:

It is not necessary that the premises and conclusion are related to each other. On the basis of the formulation of P and Q, the interpretation of the truth table is dependent.

For example:

• If Jack is made of plastic, then the Ocean is green.
• The statement: Jack is made of plastic
• The statement: The Ocean is green

The above two statements are not making any sense because Jack is a human, and he can never be made of plastic, and another statement Ocean is green will never happen because the ocean is always blue and the color of Ocean can't be changed. As we can see that both statements are not related to each other. On the other hand, the truth table for the statement P ⇒ Q is valid. So it is not a question of whether the truth table is correct or not, but it is a question of imagination and interpretation.

So in P ⇒ Q, we don't need any type of connection between the premise and consequent. On the basis of the true value of P and Q, the meaning of these only depends.

These statements will also be false even if we consider both the statements for our world, so

So when we look at the above truth table, we see that when P is false and Q is false, then P ⇒ Q is true.

So, if the Jack is made of plastic, then the Ocean will be green.

However, premise p and conclusion q will be related, and both statements make sense.

Ambiguity

There can be an ambiguity in the implied operator. So when we use the imply operator (⇒), at this time, we should use the parenthesis.

For example: In this example, we have an ambiguous statement P ⇒ Q ⇒ R. Now, we have two ambiguous statements ((P ⇒ Q) ⇒ R) or (P ⇒ (Q ⇒ R)), and we have to show whether these statements are similar or not.

Solution: We will prove this with the help of a truth table, which is described as follows:

In the above truth table, we can see that the truth table of P ⇒ (Q ⇒ R) and (P ⇒ Q) ⇒ R are not similar. Hence, they both will generate different outputs or results.

More about Implication

Some more examples of Implications are described as follows:

• If it is sunny, then I will go to school.
• If I get a good job, then I will earn money.
• If I get good marks, then my parents will be happy.

In all the above examples, we get confused because we don't know when an implication will be considered as true and when it will be considered as false. To solve this problem and to understand the concept of implication, we will use a hypothetical example. In this example, we will assume that Marry will play badminton with his boyfriend Jack, and his boyfriend Jack wants to motivate Marry a little bit, so he entices her with a statement:

Through this statement, Jack means that If marry wins, then obviously he will buy a ring. Through this statement, Jack only commits himself when Marry wins. He did not commit anything in any case when Mary loose. So at the end of the match, there can be only four possibilities, which are described as follows:

• Marry wins - buy a ring.
• Marry wins - buy no ring.
• Marry loses - buy a ring.
• Marry loses - buy no ring.

However, Jack did not make any statement related to rule (B). He also did not mention rules number (C) and (D) in his statement, so if Marry loose, then it's totally up to Jack to buy a ring for her or not. In effect, statements (A), (C), and (D) might happen as the outcome of the statement that Jack says to Marry, but (B) will not be the outcome. If outcome (B) occurs, only then Jack will be caught in a lie. In all the other three cases, i.e., (A), (C), and (D), he will have spoken the truth.

Now we will use the simpler statement so that we can symbolically define Jack's statement like this:

In this implication, we use the logical symbol ⇒, which can be read as "implies". We will form the Jack's Compound statement with the help of putting this arrow from P to Q like this:

In conclusion, we have observed that the implication will be false only when P is true and q is false. According to this statement, Marry wins the game, but sadly Jack does not buy a ring. In all the other cases/outcomes, the statement will be true. Accordingly, the truth table for implication is described as follows:

The list of corresponding logic equations for the implication is described as follows:

Examples of Implication:

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

Example 1: Suppose there are four statements, P, Q, R, and S where

P: Jack is in school

Q: Jack is teaching

R: Jack is sleeping

S: Jack is sick

Now we will describe some symbolic statements which are involved with these simple statements.

1. P → R
2. S → ~P
3. ~Q → (S ∧ R)
4. (P ∨ R) → ~Q
5. (~R ∧ ~S) → (Q ∨ ~P)

Here we have to show the representation of interpretation of these symbolic statements into words.

Solution:

Example 2: In this example, we have an implication P → Q. Here, we also have three more compound statements that are naturally associated with this implication that is contra positive, inverse, and converse of the implication. The relation between all these four statements is described with the help of a table, which is described as follows:

Now we will consider an example of implication, which has the statement, "If you study well, you get good marks". This statement is in the form P → Q, where

P: you study well

Q: you get good marks

Now we will use the P and Q statements and show the four associate statements like this:

Implication: If you study well, you get good marks.

Converse: If you get good marks, you study well.

Inverse: If you do not study well, you do not get good marks.

Contrapositive: If you do not get good marks, you do not study well.

The truth values of all the above associate statements are described with the help of a truth table, which is described as follows

In the above table, we can see that the implication (P → Q) and its contrapositive (~Q → ~P) have the same value in their columns. That means they both are equivalent. So we can say that:

Similarly, we can see that the converse and inverse both have similar values in their columns. But this will not make any difference because the inverse is the contra-positive of the converse. Similarly, the original implication can get from the contra-positive of the contra-positive. (That means if we negate P and Q and then switch the arrow's direction, and after that, we will again repeat the process, that means negate ~P and ~Q, and again switch the arrow's direction, in this case, we will get back where we started).