Section 1.7 and 1.8- Deductive Structure / Statements of Logic Advanced Geometry Section 1.7 and 1.8- Deductive Structure / Statements of Logic Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. 1 A B C D E
Building blocks of Geometry: Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. Building blocks of Geometry: Undefined Terms - Terms which we can't officially define. We describe their properties but don't formally define them. examples: point, line, plane Definitions - State the meaning of a term. There is never a need to prove a definition. (It is true because we say it is true.) examples: Def. of a right angle, Def. of a segment bisector...
Building blocks of Geometry (cont.): Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. Building blocks of Geometry (cont.): Postulates - An unproved assumption. These are pretty obvious statements which we are unable to prove but which clearly must be true. example: Two points determine a line. Theorems - A mathematical statement which can be proved. example: If two angles are both right angles, then they are congruent.
If two angles are both right angles, Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. Definitions, Postulates and Theorems are all conditional statements. They have some (sufficient) condition that leads to a (necessary) conclusion. If two angles are both right angles, Having two right angles is sufficient evidence to conclude that the angles are congruent. then they are congruent. When two angles are both right angles, it is necessary to conclude that they are congruent (they have to be). What happens if we reverse the condition and conclusion?
Here is the theorem with the condition and conclusion reversed: Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. Here is the theorem with the condition and conclusion reversed: If two angles are congruent Is knowing that two angles are congruent sufficient evidence to conclude that they must be right angles? then they are both right angles. If two angles are congruent, is it necessary to conclude that they are right angles? Is this a true statement? For a statement to be considered true, it must ALWAYS be true. NOT ALWAYS TRUE = FALSE
"If p then q" can be symbolized Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. Conditional statements can be written in "If p then q" form where p is the condition or hypothesis of the statement (trigger) and q is the conclusion (result). "If p then q" can be symbolized (also read "p implies q")
Definitions are always reversible. Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. Reversing the condition and the conclusion results in the Converse of the original statement. Thus the convers is symbolized If both the conditional and the converse are true, the statement is said to be reversible. Definitions are always reversible. Postulates and Theorems are sometimes reversible.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. The Negation (opposite) of the statement p is "not p" and is symbolized The Inverse of the conditional statement is formed when we negate both p and q. Thus the inverse is symbolized
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. The Contrapositive of the conditional statement is formed by both reversing and negating p and q. Thus, the contrapositive can be symbolized
Conditional T F Converse Inverse Contrapositive Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. The conditional statement and it's contrapositive have the same "truth value". That is, they are either both true or both false. The inverse and the converse have the same truth value. Conditional T F Converse Inverse Contrapositive
If two angles _______________ then ______________________ Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. Example Conditional: If two angles are both right angles, then they are congruent Converse: If two angles _______________ then ______________________ Inverse: If two angles _______________ then ______________________ Contrapositive: If two angles _______________ then ______________________
Converse: Contrapositive: Example Conditional: Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. Example Conditional: If an angle is an acute angle, then its measure is greater than 0 and less than 90. Converse: If an angle _____________________________, then __________________________________. Inverse: If an angle _____________________________, then __________________________________. Contrapositive: If an angle _____________________________, then __________________________________.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. A series of conditional statements can be connected together using the Chain Rule. If and then
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. Example: , , , and
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. At Hilldale High School, there is a rule that any student caught fighting must be given a three day suspension. Bill, Bob and Bo are all students at Hilldale. Which of the following statements is/are true? Bill has been given a three day suspension so we know that he must have been caught fighting. Bob has never been caught fighting, so we know that he has never been given a three day suspension. Bo has never been given a three day suspension, so we know that he has never been caught fighting.
If fighting then suspension. Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. At Hilldale High School, there is a rule that any student caught fighting must be given a three day suspension. Bill, Bob and Bo are all students at Hilldale. Which of the following statements is/are true? Conditional: If fighting then suspension. Bill has been given a three day suspension so we know that he must have been caught fighting. Converse: If suspension then fighting False It is not necessary for someone to Fight in order to receive a suspension. Bill could have been suspended for something else. The converse is false.
If fighting then suspension. Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. At Hilldale High School, there is a rule that any student caught fighting must be given a three day suspension. Bill, Bob and Bo are all students at Hilldale. Which of the following statements is/are true? Conditional: If fighting then suspension. Bob has never been caught fighting, so we know that he has never been given a three day suspension. Inverse: If NOT fighting then NO suspension. False Fighting is a sufficient condition to receive a suspension but not a necessary one. Bob could have been suspended for something else. The inverse is false.
If fighting then suspension. Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions. At Hilldale High School, there is a rule that any student caught fighting must be given a three day suspension. Bill, Bob and Bo are all students at Hilldale. Which of the following statements is/are true? Conditional: If fighting then suspension. Bo has never been given a three day suspension, so we know that he has never been caught fighting. If NO suspension then NOT fighting. Contrapositive: True Since suspension is a necessary result of fighting, NOT being suspended is sufficient to conclude that there was NO fighting. The contrapositive is true.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.
Learner Objective: Students will write the converse, inverse, and contrapositive of a conditional statement, determine the truth value of statements and will use the chain rule to make logical conclusions.