Geometry/Chapter 2/Lesson 3

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Introduction[edit]

In this lesson, we will be going over the following:

  • The contrapositive of a conditional statement
  • What a biconditional statement is
  • Logic symbols

Contrapositive statements[edit]

If it quacks, it is a duck.

We have already learned about conditional statements (or topics related to them) in lessons 1 and 2. We have learned, so far, about the converse and the inverse of any given conditional statement. There is one more form of conditional statements we need to learn in order for you to be able to write all of the if-then forms in Geometry. The last if-then form we have not learned yet, but will be learning today, is the contrapositive if-then form.

The contrapositive of any statement is the flipped and negated version of the original statement. This means we have to switch the conclusion for the hypothesis (and vice versa), and then negate (add not or take away not).

Examples[edit]

Statement: If it quacks, it is a duck
Contrapositive: If it is not a duck, then it does not quack.
Statement: If it is Rome, then that is in Italy.
Contrapositive: If that is not in Italy, then it is not Rome.
Statement: If the mouse does not fall, then the circus will be over.
Contrapositive: If the circus will not be over, then the mouse does fall.

Biconditional Statements[edit]

A biconditional statement is a statement that contains the phrase "if and only if".

Biconditional statement: It is a right angle if and only if it is 90 degrees.
Conditional: If it is a right angle, then it is 90 degrees.
Converse: If it is 90 degrees, then it is a right angle.
Inverse: If it is not a right angle, then it is not 90 degrees.
Contrapositive: If it is not 90 degrees, then it is not a right angle.

Logic Symbols[edit]

Conditional statements can be written up as symbols. You might be quizzed by your teacher using these logic symbols.

Term Symbol
Hypothesis p
Conclusion q
Hypothesis and Conclusion p and q
If, Then
Not ~
And ^
Or V
Therefore
If and only if (biconditional)

Example[edit]

Statement: p → q
Converse: q → p
Inverse: ~p → ~q
Contrapositive: ~q → ~p