Geometry

1. Geometry Logic

2. Today’s Agenda • Inductive reasoning • Counterexamples • Conditional Statements • Inverse • Converse • Contrapositive • Venn diagrams

3. Standards: • Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. • Find the converse, inverse, and contrapositive of a statement.

4. Inductive Reasoning • Inductive reasoning is the process of using examples and observations to reach a conclusion. • Any time you use a pattern to predict what will come next, you are using inductive reasoning. • A conclusion based on inductive reasoning is called a conjecture.

5. Counterexamples • A conjecture is either true all of the time, or it is false. • If we wish to demonstrate that a conjecture is true all the time, we need to prove it through deductive reasoning. • We will have more on deductive reasoning and the proof process later. But for now, know that we can never prove an idea by offering examples that support the idea. • However, it can be easy to demonstrate that a conjecture is false. We simply need to provide a counterexample.

6. Intro to Logic • A statement is a sentence that is either true or false (its truth value). • Logically speaking, a statement is either true or false. What are the values of these statements? • The sun is hot. • The moon is made of cheese. • A triangle has three sides. • The area of a circle is 2πr. • Statements can be joined together in various ways to make new statements.

7. Conditional Statements • A conditional (or propositional) statement has two parts: • A hypothesis (or condition, or premise) • A conclusion (or result) • Many conditional statements are in “If… then…” form. • Ex.: If it is raining outside, then I will get wet. • A conditional statement is made of two separate statements; each part has a truth value. But the overall statement has a separate truth value. What are the values of the following statements? • If today is Friday, then tomorrow is Saturday. • If the sun explodes, then we can live on the moon. • If a figure has four sides, then it is a square.

8. Conditional Statements • Sometimes conditional statements aren’t in “If… then…” form. See if you can determine the hypothesis and conclusion in each of the following, and restate in “If… then…” form. • An apple a day keeps the doctor away. • What goes up must come down. • All dogs go to heaven. • Triangles have three sides.

9. Inverse • The inverse of a statement is formed by negating both its premise and conclusion. • Statement: • If I take out my cell phone, then Mr. Peterson will confiscate it. • Inverse: • If I do take out my cell phone, then Mr. Peterson will confiscate it. not not

10. Try these • Give the inverses for the following statements. (You may wish to rewrite as “If… then…” first.) Then determine the truth value of the inverse. • Barking dogs give me a headache. • If lines are parallel, they will not intersect. • I can use the Pythagorean Theorem on right triangles. • A square is a four-sided figure.

11. Converse • A statement’s converse will switch its hypothesis and conclusion. • Statement: • If I am happy, then I smile. • Converse: • If , then . I am happy I smile

12. Try these • Give the converses for the following statements. Then determine the truth value of the converse. • If I am a horse, then I have four legs. • When I’m thirsty, I drink water. • All rectangles have four right angles. • If a triangle is isosceles, then two of its sides are the same.

13. Contrapositive • A contrapositive is a combination of a converse and an inverse. The hypothesis and conclusion switch, and both are negated. • Statement: • If my alarm has gone off,then I am awake. • Contrapositive: • If ,then . my alarm has not gone off not I am not awake not

14. Try these • Give the contrapositives for the following statements. Then determine its truth value. • If it quacks, then it is a duck. • When Superman touches kryptonite, he gets sick. • If two figures are congruent, they have the same shape and size. • A pentagon has five sides. • Note: A contrapositive always has the same truth value as the original statement!

15. Symbolic representation • Logic is an area of study, related to math (and computer science and other fields). In formal logic, we can represent statements symbolically (using symbols). • Some common symbols: a statement, usually a premise a statement, usually a conclusion creates a conditional statement negates a statement (takes its opposite)

16. Examples • If p, then q • Inverse:If not p, then not q • Converse:If q, then p • ContrapositiveIf not q, then not p

17. Conjunctions • A conjunction consists of two statements connected by ‘and’. • Example: • Water is wet and the sky is blue. • Notation: • A conjunction of p and q is written as

18. Disjunctions • A disjunction consists of two statements connected by ‘or’. • Example: • I can study or I can watch TV. • Notation: • A disjunction of p and q is written as

19. Biconditional • A biconditional statement is a special type of conditional statement. It is formed by the conjunction of a statement and its converse. • Biconditional statements can be shortened by using “if and only if” (iff.). • A biconditional is true whether you read it forwards or ‘backwards’.

20. Biconditional • A good definition will consist of a biconditional statement. • Ex: A figure is a triangle if and only if it has three sides. • This means that if a figure is a triangle, then it has three sides and if a figure has three sides, then it is a triangle. It is true forwards and backwards.

21. Venn Diagrams • The truth values of compound statements can also be represented in Venn diagrams. • p: A figure is a quadrilateral. • q: A figure is convex. • Which part of the diagramrepresents: p q

22. Venn Diagrams – Conditionals • A Venn diagram can represent a conditional statement: • p: A figure is a quadrilateral. • q: A figure is a square. p q