In the study of logic, when two statements are combined then they are known as compound statements. They can be conditional or bi-conditional. If a statement is given that if it rains we will not play. This is a combination of two statements. This statement is in the form of "If ... then" and hence is a conditional statement. A conditional statement having statements p and q can be written in given ways.

1: p implies q.

2: p is sufficient for q.

3:q is necessary for p.

4: p $\Rightarrow$ q

Let us take an example and understand. a = 6 then b = a + 2 must be 8. Hence, a $\Rightarrow$ b.

Below are the listed types of conditional statements:

1. Conditional statement

2. Bi – conditional statement

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Conditional statement is also called as implications.

Form of conditional statement is if P then Q

Sign of logical connector conditional statement is $\rightarrow$. Example P $\rightarrow$ Q pronouns as P implies Q.

The state P $\rightarrow$ Q is false if the P is true and Q is false otherwise P $\rightarrow$ Q is true.

** Truth table for Conditional statement:**

A | B | A$\rightarrow$B |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

We have a conditional statement If it is raining, we will not play. Let, A: It is raining and B: we will not play.

If A is true, that is, it is raining and B is false, that is, we played, then the statement A implies B is false.

If A is false, that is, it is not raining and B is true, that is, we did not play, still the statement is true. As A is the necessary condition for B but it is not sufficient. If A is true, B should be true but if A is false B may or may not be true.

$P\leftrightarrow Q \Rightarrow (P\rightarrow Q)\vee (Q\rightarrow P)$

P: A number is divisible by 2.

Q: A number is even.

If P will occur then Q will occur and if Q will occur then P will occur.

Hence, P will occur if and only if Q will occur.

We can say that $P\leftrightarrow Q$.

**Problem 1:** If a > 0 is a positive number then a = 5 is correct or not justify your answer.

**Solution: **

**Step 2: **so it is given a=5

**Step 3:** here first thing is correct that a is positive but

a = 5 is not correct because it can be any number greater than 0.

**Problem 2:**

P | Q | P$\Rightarrow$ |

I am late | I am on time | |

I am normal | I am abnormal |

Justify P$\Rightarrow$

**Solution:** Here if statement P is correct then statement Q is not correct

and if statement Q is correct then statement P is not correct

SO here P $\Rightarrow$ Q is not correct

because both statement are not similar.

similarly in the second statement

here if statement P is correct then statement Q is not correct

here if statement Q is correct then statement P is not correct

SO here P $\Rightarrow$ Q is not correct

because both statement are not similar.

P: x is divisible by 2.

Q: x is divisible by 4.

P: x is a prime number.

Q: x is not divisible by 2.

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