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Logic

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1. Logic! Yes, Captain Kirk. That would be logical.

2. Inductive and Deductive Reasoning The two types of reasoning can be explained in a simple way. Deductive Reasoning Inductive Reasoning Using a general example of formula Using specific examples to to explain a specific happening state a general formula general to specific specific to general For instance, The general formula for finding Mary�s hen lays three eggs Monday, the area of a rectangle is length X width. Tuesday, and Wednesday. Mary What is the specific area of this room? Concludes her hen will always lay Three eggs everyday. In our geometry class we use both inductive and deductive reasoning everyday. Example Inductive: The discovery worksheets we do in class ask you to compare the group member answers and state a general theorem. (Given the base angles are congruent in all triangles what can you conclude? In one triangle, if the angels opposite the congruent sides are congruent.) Example Deductive: Knowing the theorem, if two parallel lines are cut by a transversal, then the corresponding angles are congruent, we can conclude in the picture below that 1 <1 is congruent to <2. 2

3. Special Words When deciding if the statement is inductive or deductive reasoning, look for key words which will help you. Words such as every, all, always - indicate "general" One , only, single - indicate "specific"

4. Example 1 Is this inductive or deductive? Given the equation What is the size of the third side of the triangle is a = 4 and b = 3? Deductive

5. Example 2 Fred notices in his science class he had a test on Tuesday the first week of school. During the second and third week, he also had a quiz on Tuesday. Fred concludes he will have a quiz every Tuesday. Inductive

6. Conditional Statements A conditional statement is a sentence written in the if-then form. Example I: If it is raining then there are clouds in the sky. Example II: If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

7. Example 3 If two sides of a triangle are congruent, then the opposite angles of the triangle are congruent. What is the hypothesis? two sides of a triangle are congruent What is the conclusion? The opposite angles of the triangle are congruent.

8. Parts of Conditional Statements If it is raining then there are clouds in the sky. The hypothesis is underlined once. The conclusion is underlined twice. Notice if and then are not part of the hypothesis nor the conclusion.

9. Changing the form of the Conditional Statement The symbol for writing a conditional statement is read, �If p then q� The symbol meaning �not� or �negation� is ~ .

10. Direct Statement When dealing with conditional statements, the statement you begin with is called the direct statement. Everything changes according to the direct statement. Direct Statement: If it is raining, then there are clouds in the sky. Symbol: Converse Statement: If there are clouds in the sky, then it is raining. (Notice the hypothesis and the conclusion switched places.) Symbol: Inverse Statement: If it is not raining, then there are not clouds in the sky. (Notice the hypothesis and conclusion have been negated.) Symbol: Contrapositve Statement: If there are not clouds in the sky, then it is not raining. (Notice the hypothesis and conclusion has been switched and negated.) Symbol:

11. Counterexample The one false statement which makes the direct statement false. In this case, if you could show it was raining without clouds in the sky, this would be a counterexample. The counterexample ALWAYS negates the conclusion to the statement. You use counterexamples to prove scientific labs.

12. Example 3 You are doing a science fair project. Your direct statement is, If bread sets in the sun, then it will grow mold. To prove this statement, you would have to set EVERY piece of bread in the world in the sun and make a conclusion. What is the counterexample? If bread sets in the sun, then it will NOT grow mold. Then you would only have to find one falsehood, and the experiment is over. ?

13. Biconditional Statement Definition: When the direct and the converse statements are BOTH true. Example: If a triangle is equilateral triangle, then it is equiangular. Is this true? Write the converse. If a triangle is equiangular then it is equilateral. Is this true? When both are true it is a biconditional statement.

14. Biconditional Statement If they are both true it can be put together into one statement called the biconditional statement. In this case: It is an equilateral triangle if and only if it is equiangular. If and only if means: Symbol: or sometimes written p iff q.

15. Biconditional Statement Example: The number is positive if and only if is positive. What is the direct statement? If a number is positive, then is positive. What is the converse? If is positive, ten a number is positive. Are both true?

16. Logically Equivalent When the direct statement, converse, inverse and countrapositive are ALL True or ALL False, then the statement is logically equivalent. Symbol:

17. Homework Honors: pages 5 the top only, 10 and 16 Regular: Pages 5 at the top, 6 and 9