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geometry honors section 2 2 introduction to logic and introduction to deductive reasoning
Geometry Honors Section 2.2Introduction to Logic and Introduction to Deductive Reasoning
slide2

whale

mammal

whale

mammal

The figures at the right are Venn diagrams. Venn diagrams are also called __________ diagrams, after the Swiss mathematician _______________________.Which of the two diagrams correctly represents the statement“If an animal is a whale, then it is a mammal”.

Euler

Leonard Euler

slide3

If-then statements like statement (1) are called *___________ In a conditional statement, the phrase following the word “if” is the *_________. The phrase following the word “then” is the *_________.

conditionals.

hypothesis

conclusion

slide4
If you interchange the hypothesis and the conclusion of a conditional, you get the *converseof the original conditional.
slide5

Example 1: Write a conditional statement with the hypothesis “an animal is a reptile” and the conclusion “the animal is a snake”. Is the statement true or false? If false, provide a counterexample.

If an animal is a reptile than it is a snake.

False

slide6
Write the converse of the conditional statement. Is the statement true or false? If false, provide a counterexample.

If an animal is a snake, then it is a reptile.

True

slide7

Example 2: Consider the conditional statement “If two lines are perpendicular, then they intersect to form a right angle”. Is the statement true or false? If false, provide a counterexample.

slide8
Write the converse of the conditional statement. Is the statement true or false? If false, provide a counterexample.
slide9

When an if-then statement and its converse are both true, we can combine the two statements into a single statement using the phrase “if and only if” which is often abbreviated iff.Example: Combine the two statements in example 2, into a single statement using iff.

slide10

Reasoning based on observing patterns, as we did in the first section of Unit I, is called inductivereasoning. A serious drawback with this type of reasoning is

your conclusion is not always true.

deductive reasoning is reasoning based on deductive reasoning
*Deductive reasoning is reasoning based onDeductive reasoning

logically correct conclusions

always give a correct conclusion.

slide12

We will reason deductively by doing two column proofs. In the left hand column, we will have statements which lead from the given information to the conclusion which we are proving. In the right hand column, we give a reason why each statement is true. Since we list the given information first, our first reason will always be ______. Any other reason must be a _________, _________ or ________.

given

theorem

postulate

definition

slide14

Our first proofs will be algebraic proofs. Thus, we need to review some algebraic properties. These properties, like postulates are accepted as true without proof.

addition property of equality subtraction property of equality multiplication property of equality
Addition Property of Equality:Subtraction Property of Equality:Multiplication Property of Equality:

If a = b, then a+c=b+c

If a = b, then a-c = b-c

If a = b, then ac =bc

substitution property
Substitution Property:

If two quantities are equal, then one may be substituted for the other in any equation or inequality.

example complete this proof
Example: Complete this proof:

Given

Multiplication Property

Distributive Property

Addition Property

Division Property

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