Conditional Statements Lesson 2-1. Conditional Statements have two parts: Hypothesis ( denoted by p) and Conclusion ( denoted by q)

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Presentation transcript:

Conditional Statements Lesson 2-1

Conditional Statements have two parts: Hypothesis ( denoted by p) and Conclusion ( denoted by q)

Hypothesis (p) Phrase following “if” the given information

Conclusion (q) Phrase following “then” the result of the given information

Conditional statements can be put into an “if-then” form to clarify which part is the hypothesis and which is the conclusion.

Example: Vertical angles are congruent. If two angles are vertical, then they are congruent. can be written as...

If two angles are vertical, then they are congruent. p implies q Hypothesis (p): two angles are vertical Conclusion (q): they are congruent

Conditional Statements can be true or false: A conditional statement is false only when the hypothesis is true, but the conclusion is false. A counterexample is an example used to show that a statement is not always true and therefore false.

Giving a Counterexample Therefore (  ) the statement is false. Statement: If you live in Virginia, then you live in Richmond. True or False? Give a counterexample: Henry lives in Virginia, BUT he lives in Ashland.

Symbolic Logic

Symbols can be used to modify or connect statements.

 is used to represent the word “therefore”

Example H : I watch football HH Therefore, I watch football

 is used to represent implies used in if … then

Example p: a number is prime q: a number has exactly two divisors p  q: If a number is prime, then it has exactly two divisors.

~ is used to represent the word “not” or “negate”

Example w: the angle is obtuse ~w: the angle is not obtuse Be careful because ~w means that the angle could be acute, right, or straight

Example r: I am not happy ~r: I am happy Notice: ~r took the “not” out… it would have been a double negative (not not)

 is used to represent the word “and”

Example a: a number is even b: a number is divisible by 3 a  b: A number is even and it is divisible by 3. 6,12,18,24,30,36,42...

 is used to represent the word “or”

Example a: a number is even b: a number is divisible by 3 a  b: A number is even or it is divisible by 3. 2,3,4,6,8,9,10,12,14,15,...

iff is used to represent the phrase “if and only if”

Example h: I watch football k: the Eagles are playing h iff k I watch football if and only if the Eagles are playing

Different Forms of Conditional Statements

A conditional statement can be written in three different ways. These three new conditional statements can be true or false. EXAMPLE: p  q If two angles are vertical, then they are congruent.

Converse: q  p SWITCH (p and q but not if or then) If two angles are congruent, then they are vertical.

Inverse: ~p  ~q NEGATION (keep same order) If two angles are not vertical, then they are not congruent.

Contrapositive:~ q  ~p SWITCH and NEGATE If two angles are not congruent, then they are not vertical.

Contrapositives are logically equivalent to the original conditional statement. If p  q is true, then  q  p is true. If p  q is false, then  q  p is false.

Biconditional If a conditional statement and its converse are both true, then the two statements may be combined. using the phrase if and only if (iff)

Definitions are always biconditional Statement: If an angle is right then it has a measure of 90 . Converse: If an angle measures 90 , then it is a right angle. Biconditional: An angle is right iff it measures 90 