**Negation, Converse &
Inverse | Truth Table For Conditional Statemen****ts**

In conditional statements, "If **p** then **q**" is denoted
symbolically by "**p q**";
**p** is called the **hypothesis** and **q**
is called the **conclusion**. For instance,
consider the two following statements:

**If Sally passes the exam, then she will get
the job.**

**If 144 is divisible by 12, 144 is divisible
by 3.**

Let** p** stand for the statements "Sally passes the exam" and
"144 is divisible by 12".

Let **q **stand for the statements "Sally will get the job" and
"144 is divisible by 3".

The hypothesis in the first statement is "144 is divisible by 12", and the conclusion is "144 is divisible by 3".

The second statement states that Sally will get the job**
if **a certain condition (passing the exam) is met; it says nothing about
what will happen if the condition is **not **met. If
the condition is not met, the truth of the conclusion cannot be determined;
the conditional statement is therefore considered to be **vacuously true**,
or **true by default**.

Let **p** and **q** be statement variables which
apply to the following definitions.

In expressions that include and
other logical operators such as ,
, and ~, the **order
of operations** is that is
performed last while ~ is performed first.

**Representation
of If-Then as Or***
*Let ~

p q ~p q |

The negation of a conditional statement is represented symbolically as follows:

~(p q) p ~q |

By definition, **p q**
is false if, and only if, its hypothesis, **p**, is true and its conclusion,
**q**, is false.

The **converse**
and **inverse**
of a conditional statement are logically equivalent
to each other, but neither of them are logically equivalent to the conditional
statement

Practice Exercises |

**Truth
Table For Conditional Statements**

p |
q | p q | q p | p q | (p q) (q p) |

T | T | T | T | T | T |

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

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

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