2BiconditionalThe Biconditional is a compound statement that combines 2 conditionals and the connector “and”(p q) (q p).That is (p implies q) and (q implies p).It is written as p q.A biconditional is called an if and only if statement.
3A biconditional statement is only true when both p and q have the same truth values. p qTF
4Two statements are logically equivalent if each statement has the same truth values. To show this we us the logical connectorif and only if.In symbolic form if and only if is represented by the symbol: Example:Prove that pq is logically equivalent to~pq
6Inverse, Converse and Contrapositives In conducting an argument the conditional statement is the one we use most often.In order to use them correctly we must understand their different forms and how they are related.
7The InverseThe inverse is formed by negating both the hypothesis and the conclusion.ExampleIf a number is divisible by 4 then it is also divisible by 2.The inverse:If a number is not divisible by 4 then it is not divisible by 2.
8A true conditional can have a false inverse (as seen in the last example).Sometimes a false conditional can have a true inverse.Sometimes both the statement and the inverse have the same truth values.
9In symbolic form The inverse written in symbolic form Examples p q is the statement~p ~q is the inverse.or~p q is the statement then p ~q is the inverse.
10The ConverseThe converse is formed by interchanging both the hypothesis and the conclusion.ExampleIf a polygon has four sides then it is a quadrilateral. (the statement)If a polygon is a quadrilateral then it has four sides. (the converse)
11Like the inverse a conditional statement can have a false converse, a true converse or they can have the same truth values.In symbolical form:pq is the statement q p is the converse.
12The Contrapositive.The contrapositive is formed by doing an inverse followed by a converse or a converse followed by an inverse.
13The conditional statement and it’s contrapositive are equivalent statements. pq is logically equivalent to~q ~p