1. The System of Mathematics Logic: Negation, Inverses, Contrapositives, and Logical Equivalence
Geometry – Mr. Ferguson

2. Negation Not all mathematical statements are affirmative:
If a function is linear, then it is not quadratic.
If your counselor is not Mr. Ornelas, then it is Mr. Williams.
If you are not a sophomore, then you are not taking Geometry.
Negative hypotheses or conclusions assert ideas that are not true or not false.
In order to address the negative statements, we introduce the notion of negation.  

3. Negation A negation “negates” the truth value of a statement.
If a statement is true, then the negation of that statement makes it false. 
If a statement is false, then the negation of that statement is true. 
Negation symbol:  ~
Example:  P = two angles are adjacent; 
~P = two angles are not adjacent
In logic, you CAN have double negatives (or more!): 
~(~P) will always have the same truth value as P 

4. If ~[Hypothesis], then ~[Conclusion]
Inverse
In order to form the inverse of a conditional, you negate both the hypothesis and the conclusion. 
If ~[Hypothesis], then ~[Conclusion]
Conditional:  If x is an even number,  then x2  is an even number.
Inverse:  If x is not an even number, then  x2  is not an even number.
Converse:  If x2 is an even number, then x is an even number.
This statement is called the inverse because you “invert” the truth values of both the hypothesis and conclusion. 

5. If ~[Conclusion], then ~[Hypothesis]
Contrapositive
In order to form the contrapositive of a conditional, you negate the truth values of both the hypothesis and conclusion, AND you swap their positions. 
If ~[Conclusion], then ~[Hypothesis]
Conditional:  If a quadrilateral has 4 right angles, then it is a rectangle.
Contrapositive:  If a quadrilateral is not a rectangle, then it does not have four right angles. 

6. All right angles are congruent.
Example
Write the conditional, converse, inverse, and contrapositive of the following statement:
All right angles are congruent.

7. Logical Equivalence
Two statements are logically equivalent if and only if their biconditional is always true. 
That is, it is IMPOSSIBLE for one statement to be true and the other NOT be true. Logically equivalent statements always have the same truth value!
Conditional:   If n2 is odd, then n is odd. 
Contrapositive:  
Are the conditional and the contrapositive logically equivalent in this case? 

8. Logical Equivalence
Conditional and Contrapositive are logically equivalent.
Inverse and Converse are logically equivalent.
Conditional and Converse are NOT always truth- equivalent.