Conditional Statements and Deductive Reasoning 2-1 – 2-3

Conditional Statements A conditional statement is also known as an if-then statement. If you are not completely satisfied then your money will be refunded. Every conditional has two parts, the part following the if, called the hypothesis, and the part following then, which is the conclusion. If Texas won the 2006 Rose Bowl game, then Texas was college football’s 2005 national champion. Hypothesis – Texas won the 2006 Rose Bowl game. Conclusion – Texas was college football’s 2005 national champion.

Conditional Statements This also works with mathematical statements. If T – 38 = 3, then T = 41 Hypothesis: T – 38 = 3 Conclusion: T = 41

Conditional Statements Sometimes the “then” is not written, but it is still understood: If someone throws a brick at me, I can catch it and throw it back. Hypothesis: Someone throws a brick at me Conclusion: I can catch it and throw it back.

Name the hypothesis and the conclusion  If you want to be fit, then get plenty of exercise.  If you can see the magic in a fairy tale, you can face the future.  “If someone throws a brick at me, I can catch it and throw it back.” – Harry S. Truman  “If you can accept a deal and open your pay envelope without feeling guilty, you’re stealing.” – George Allen, NFL Coach  “If my fans think I can do everything I say I can do, then they are crazier than I am.” – Muhammed Ali

Writing a Conditional You can write many sentences as conditionals.  A rectangle has four sides.  If a figure is a rectangle, then it has four sides.  A tiger is an animal.  If something is a tiger, then it is an animal.  An integer that ends with 0 is divisible by 5.  A square has four congruent sides.

Truth Value  A conditional has a truth value of either true or false. To show a conditional is true, show that every time the hypothesis is true, then the conclusion is true. If you can find one counterexample where the hypothesis is true, but the conclusion is false, then the conditional is considered false.

Counterexamples  To show that a conditional is false, find one counterexample. If it is February, then there are only 28 days in the month. This is typically true, however, because during leap years February has 29 days, this makes the conditional false. If a state begins with New, then the state borders an ocean. This is true of New York, and New Hampshire, but New Mexico has no borders on the water, therefore this is a false conditional.

Using a Venn Diagram If you live in Chicago, then you live in Illinois. Residents of Chicago Residents of Illinois

Converse The converse of a conditional switches the hypothesis and the conclusion. Conditional: If two lines intersect to form right angles, then they are perpendicular. Converse: If two lines are perpendicular, then they intersect to form right angles If two lines are not parallel and do not intersect, then they are skew.

Finding the Truth of a Converse It is possible for a conditional and its converse to have two different truth values. If an animal is a chicken, then it is covered in feathers. If an animal is covered in feathers, then it is a chicken. Although the first conditional is true, as all chickens have feathers, the second is false, as chickens are not the only animal with feathers.

Finding the Truth of a Converse

Biconditionals  When a conditional and its converse are true, you can combine them as a true biconditional. You do this by connecting the two parts of the conditional with the words “if and only if” If two angles have the same measure, then the angles are congruent. If two angles are congruent, then the angles have the same measure.  Since both of these are true, the biconditional statement would read: Two angles are congruent if and only if the angles have the same measure.

Separating a Biconditional into Parts  You can also rewrite a biconditional as a conditional and its converse. A number is divisible by 3 if and only if the sum of its digits is divisible by 3.  This would be separated into the two statements: If a number is divisible by 3, then the sum of its digits is divisible by 3. If the sum of a number’s digits is divisible by 3, then it is divisible by 3.

Definitions  A good definition has several components:  It uses clearly understood terms.  It is precise and specific. (avoid opinion and use of words such as large, sort of, and almost).  It is reversible. This means that a good definition can be written as a true biconditional. Definition of Perpendicular Lines: Perpendicular Lines are two lines that intersect to form right angles. Conditional: If two lines are perpendicular, then they intersect to form right angles. Converse: If two lines intersect to form right angles, then they are perpendicular.  Since the two are both true, the definition can be written as a true biconditional: Two lines are perpendicular if and only if they intersect to form right angles.

Chapter 2.3 Deductive Reasoning

Vocabulary  Deductive Reasoning (logical reasoning) = the process of reasoning logically from given statements to a conclusion. You conclude from facts.

Example  A physician diagnosing a patient’s illness uses deductive reasoning. They gather the facts before concluding what illness it is.

Law of Detachment  Law of Detachment = If a conditional is true and its hypothesis is true, then its conclusion is true  In symbolic form:  If p  q is a true statement and p is true, then q is true

Law of Detachment Example  Use Law of Detachment to draw a conclusion If a student gets an A on a final exam, then the student will pass the course. Felicia gets an A on the music theory final exam Conclusion: She will pass Music Theory class

Law of Detachment: Try it  Use Law of Detachment to draw a conclusion If it’s snowing, then the temperature is below 0 degrees It is snowing outside Conclusion: It’s below 0 degrees

Law of Detachment Example: Careful!  Use the Law of Detachment to draw a conclusion If a road is icy, then driving conditions are hazardous Driving conditions are hazardous Conclusion: You can’t conclude anything!!!

Quick Summary  You can only conclude if you are given the hypothesis!!!  If you are given the conclusion you CAN NOT prove the hypothesis

Law of Syllogism  If p  q and q  r are true statements, then p  r is a true statement  Example: If the electric power is cut, then the refrigerator does not work. If the refrigerator does not work, then the food is spoiled. So if the electric power is cut, then the food is spoiled.

Law of Syllogism example:  Use law of syllogism to draw a conclusion  If you are studying botany, then you are studying biology.  If you are studying biology, then you are studying a science  Conclusion: If you are studying botany, then you are studying a science

Law of Syllogism: Try it  Use the law of Syllogism and draw a conclusion  If you play video games, then you don’t do your homework  If you don’t do your homework, then your grade will drop  Conclusion: If you play video games, then your grade will drop

Homework  Page Complete even  Page 90 Complete 2-22 even  Page 96 #2, 4, 8, 10, 11, 12, 14  Due Tomorrow