 What are conditionals & biconditionals?  How do you write converses, inverses, and contrapositives?

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Presentation transcript:

 What are conditionals & biconditionals?  How do you write converses, inverses, and contrapositives?

 Has two parts, a hypothesis and conclusion  If, then statements  If  hypothesis  Then  conclusion  Example: If we all get our work done, then there will be no homework.

 Share with the person next to you

 All dogs bark.  Two angles that are a vertical pair have the same angle measure.  The measure of a straight angle is 180.

 In logic, conditional statements are represented by symbols and variables.  The statement: If it is raining, then there are clouds in the sky.  Let R = it is raining, and Let C = there are clouds in the sky.  The logic notation would be written:  R  C

 #3 – 5,

 The opposite of the original statement  Adds in a “not”  The negation of the sentence:  If it is raining, then there are clouds in the sky.  If it is not raining, then there are not clouds in the sky.

 The car was expensive.  The game was not broken.

 When the hypothesis and conclusion are switched this is called the converse.  The converse takes on a different meaning from the original statement and is not always correct.  When we took the converse of that statement, the statement was no longer true. There can be clouds and no rain.

 If I get an A on the test, then my parents will take me out to eat.  If my parents took me out to eat, then I got an A on my test.  Is it true?  Can you think of a statement that the converse would be true?

 The inverse is when we take the negation of the conditional statement.  If it is raining, then there are clouds in the sky.  Inverse:  If it is not raining, then there are not clouds in the sky.  Is this true?

 If you beat me in the game, then I’ll scream.  If you don’t beat me in the game, then I won’t scream.  Is it true?

 Negate the statement and switch the hypothesis and conclusion.  If you are a guitar player, then you are a musician.  If you are not a musician, then you are not a guitar player.  Is it true?

 Original (P = Hypothesis, Q = Conclusion)  P  Q  Converse  Q  P  Inverse  ~P  ~Q  Contrapositive  ~Q  ~P

 Pairs of statements that take on the same meaning.  The original conditional statement and the contra positive are equivalent statements because they take on the same meaning.  The converse and inverse are equivalent statements also.

 Original Statement  If it is raining, then there are clouds in the sky  Contrapositive  If there are no clouds in the sky, then it is not raining.  They take on the same meaning.

 If and only if statements  These statements are considered true when  The hypothesis and conclusion are both true.  The hypothesis and conclusion are both false.  Notation: P Q

 Two lines are perpendicular if and only if they intersect to form a right angle.  An angle is considered obtuse if and only if the angle measure is between 90 and 180

 Write the following as a biconditional statement  Coplanar points are points that lie in the same plane.