This lesson explains the concepts, how to write and application of the Biconditional,
Inverse, Converse, and Contrapositives. The content, explanations and including
practice problems with solution can be learnt using video audio presentation in
own hand writing by the instructor and several examples and will help you to apply
the developed skills for finding solution to real-world problems.
The biconditional consists of two conditionals and it is a compound sentence formed by the conjunction of the two conditionals pq
(read as, if and only if). Simply saying: a biconditional statement includes a condition and its converse. Both must be true. E.g. "If the alternate interior angles are equal, then the lines are parallel" is true and also its converse is true. Thus it is a biconditional statement. Biconditionals can be used to to write definitions, to solve the equations and to state the equivalences.
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Remember: The biconditional p if and only if q is true when p and q are both true and both false. It can also said, pq
is true when p and q have the ‘same truth value’. If p and q have different truth values, the biconditional is false.
For example- Examine whether it is a biconditional: “A m135º is an obtuse angle”.
If you look into it, it is not reversible i.e. its converse is false.
The converse would read as ‘An obtuse angle’s measure is 135º. However,
you know that obtuse angle can have any measurements more than 90º, but
less than 180º. Since the converse is not true, hence it is not case of
Inverse, Converse, and Contrapositives Inverses:
Generally the conditional if p then q is the connective most often used in reasoning. However; with some changes in words in the original statement, additional conditionals can be formed. These new conditionals are called the inverse, the
converse, and the contrapositive.
Inverse is a statement formed by negating the hypothesis and conclusion of the original conditional. Symbolically, the inverse is written as (~p
~q). The symbols for the inverse may be read as: not p, implies
not q OR if not p, then not q. E.g. Right angle is defined as- an angle whose measure is 90 degrees. If you are to write it as inverse statement, it can be done like: If an angle is not a right angle, then it does not measure 90.
Converse is a statement formed by interchanging the hypothesis and the conclusion
i.e. original conditional (pq)
is written as (qp).
Notice that the symbols for converse may be read as ‘q implies p’ or ‘if q, then p’. E.g. If you are to write the converse of: "If two lines don't intersect, then they are parallel", it can be written as "If two lines are parallel, then they don't intersect." It may be noted that the converse of a definition, must always be true. If this is not the case, then the definition is not valid. The converse is; therefore, can be taken as a helping tool in determining the validity of a definition.
Remember: a conditional (pq)
and its converse (qp)
may or may not be true. It is important that the truth value of each converse is judged on its own merits.
Contrapositive is a statement formed by negating both the hypothesis and conclusion
and also then interchanging these negations (~ q
~p). The symbols for contrapositive may be read as ‘not q implies not p’ or ‘if not q, then not p’. The contrapositive of a conditional statement always has the same truth value as the original statement. Therefore, the contrapositive of a definition is always true. E.g. the statement ‘A triangle is a three-sided polygon’ is true; its contrapositive, ‘A polygon with greater or less than three sides is not a triangle’ is true too.
Remember: a conditional (pq)
and its contrapositive (~ q
~p) must have the same truth value. When a conditional is true, it's contrapositive is also true and when a conditional is false, it's contrapositive is also false.
Logical equivalents: you know that have the conditional and its contrapositive have same truth value, they are termed as logical equivalents. Moreover the converse and inverse of a conditional has the relationship, such that one is the contrapositive of the other. Recall facts from earlier learning - conditional: if p then q, converse: if q then p, inverse: if not p then not q and contrapositive: if not q then not p. For example:The conditional statement is- A triangle is a polygon. It is actually an implication and if-then statement is, ‘if an object is a triangle then it is a polygon’. The converse, inverse, and contrapositive of this are:
converse: if an object is a polygon then it is a triangle (false), as not all polygons are triangles.
inverse: if an object is not a triangle then it is not a polygon (false). contrapositive: if an object is not a polygon, then it is not a triangle (true). This being true, cannot be equivalent to the converse, which is false.
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