Generally, Conditional statements are the **if-then** statement in which **p is called a hypothesis(or antecedent or premise) and q is called a conclusion( or consequence). Conditional Statements** symbolized by p, q. A Conditional statement p -> q is false when p is true and q is false, and true otherwise.

### What are propositions?

A proposition is a declarative statement that is either true or false, but not both.

**Examples:**

- Delhi is the capital of India
- 1 + 1 = 2
- 2 + 2 = 4

Let p and q are propositions.

- The conditional statement p -> q is the proposition “if p, then q”.
- The conditional statement p -> q is false when p is true and q is false and true in all other cases.

**By the following table, we can identify the values of implications:**

p |
q |
p -> q |
---|---|---|

T |
T |
T |

T |
F |
F |

F |
T |
T |

F |
F |
T |

### Variety of terminology is used to express p -> q

**” if p then q****“if p, q “****“q if p”****“q when p”****“q unless p”****“p implies q”****“p only if q”****“q whenever p”****“q follows from p”**

Conditional statements are also called implications. The statement is an implication p -> q is called its hypothesis, and q the conclusion.

**Example: Let p be the statement “Maria learn Java Programming ” and q is the statement “Maria will find a good job”. Express the statement p -> q as a statement in English?**

**Solution: **

“If Maria learns java programming, then she will find a good job”.

or

“Maria will find a good job when she learns java programming.”

**Converse, Contrapositive, and Inverse**

We can form some new conditional statements starting with a conditional statement p -> q.

- The
**converse of p -> q**is the proposition**q -> p**. - The
**contrapositive of p -> q**is the proposition**~q -> ~p**. - The
**inverse of p -> q**is the proposition**~p -> ~q**.

**By the following table, we can identify the values of Converse, Contrapositive, and Inverse:**

p |
q |
~p |
~q |
p -> q |
~q -> ~p |
---|---|---|---|---|---|

T |
T |
F |
F |
T |
T |

T |
F |
F |
T |
F |
F |

F |
T |
T |
F |
T |
T |

F |
F |
T |
T |
T |
T |

Note:The contrapositive always has the same truth value as p -> q. When two compound propositions always have the same truth value we call them equivalent, so conditional statement and its contrapositive are equivalent. The converse and the inverse of a conditional statement are also equivalent.

**Example 1: Show that p -> q and its contrapositive ~q -> ~p are logically equivalent.**

**Solution:**

p |
q |
~p |
~q |
p -> q |
~q -> ~p |
---|---|---|---|---|---|

T |
T |
F |
F |
T |
T |

T |
F |
F |
T |
F |
F |

F |
T |
T |
F |
T |
T |

F |
F |
T |
T |
T |
T |

As p ->q is equal to ~q -> ~p, hence both propositions are equivalent.

**Example 2: Show that proposition q -> p, and ~p -> ~q is not equivalent to p -> q.**

**Solution:**

p |
q |
~p |
~q |
p -> q |
q -> p |
~p -> ~q |
---|---|---|---|---|---|---|

T |
T |
F |
F |
T |
T |
T |

T |
F |
F |
T |
F |
T |
T |

F |
T |
T |
F |
T |
F |
F |

F |
F |
T |
T |
T |
T |
T |

In this case, p -> q is not equal to q -> p and ~p -> ~q, hence they are not equal to p -> q but they themselves are equal.

**Example 3: What is contrapositive, the converse, and the inverse of the conditional statement “The home team wins whenever it is raining.”?**

**Solution:**

Because “q whenever p” is one way to express conditional statements p -> q.

Original sentence:

“If it is raining, then the home team wins”.

Contrapositive:“If the home team does not win, then it is not raining.”Converse:“If the home team wins, then it is raining.”Inverse:“If it is not raining, then the home team does not win.”

**Example 4: What are contrapositive, the converse, and the inverse of the conditional statement “If the picture is a triangle, then it has three sides.”?**

**Solution:**

Contrapositive:“If the picture doesn’t have three sides, then it is not a triangle.”Converse:“If the picture has three sides, then it is a triangle.”Inverse:“If the picture is not a triangle, then it doesn’t have three sides.”

### Biconditional or Equivalence

- We now introduce another way to combine propositions that express that two propositions have the same truth values.
- Let p and q be propositions.
- The biconditional statement p <-> q is the propositions “p if and only if q”
- The biconditional statement p <-> q is true when p and q have the same truth values and is false otherwise.
- Biconditional statements are also called bi-implications.
- There are some common way to express p<->q
**“p is necessary and sufficient for q”****“if p then q, and conversely”****“p if q”.**

**By the following table, we can identify the values of Biconditional:**

p |
q |
p <-> q |
---|---|---|

T |
T |
T |

T |
F |
F |

F |
T |
F |

F |
F |
T |

**Example: What is the Biconditional of these following sentences.** **Let p be the statement” You can take the flight” and let q be the statement “You buy a ticket.”**

**Solution:**

p <-> q is “You can take the flight if and only if you buy a ticket”