2.2 Definitions and Biconditional Statements n Perpendicular Lines ­ Two lines that intersect to form right angles. m m | n A line perpendicular to a plane ­ A line that intersects a plane at a point and is perpendicular to every line in the plane. All definitions can be read as a conditional statement. They can be read forward (conditional) and backward (its converse). ex) perpendicular lines ­ 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. If a conditional and its converse are both true, we can write it as a Biconditional using if and only if. Two lines are perpendicular if and only if (IFF) they intersect to form right angles Biconditional ­ Two lines intersect iff their intersection is exactly one point. Conditional ­ If two lines intersect, then the intersection is one point. Converse ­ If two lines contain one point, then they intersect. Only if is also considered a biconditional statement. It is Saturday, only if I am working at the restaurant. Conditional ­ If it is Saturday, then I am working at the restaurant. Converse ­ If I am working at the restaurant, then it is Saturday. 1 Consider the following statement: x = 3 iff x 2 = 9 a) Is this a biconditional statement? b) Is the statement true? a) Statement is written as if and only if. It is a biconditional. b) Statement can be written as conditional: If x = 3, then x 2 = 9 (true) Converse: If x 2 = 9, then x = 3 (false, x could also be ­3) Therefore, the biconditional statement is false. 2 2.3 Deductive Reasoning ­ Laws of Logic and Symbolic Notation Symbolic Notation in a conditional is like using a variable in Algebra Conditional: If the sun is out, then the weather is good. Let p represent the hypothesis and q represent the conclusion. If p, then q. p → q Read as p implies q. (Of if p, then q) Converse: If the weather is good, then the sun is out. If q, then p. q → p Biconditional: p ↔ q Ex) Let p be "the value of x is ­5" and q be "the absolute value of x is 5" Write p → q in words Write q → p in words Is the conditional p ↔ q true? If the value of x is ­5, then the absolute value of x is 5. Conditional If the absolute value of x is 5, then the value of x is ­5. Converse The conditional is true, but the converse is false, so the biconditional is false. 3 To write an inverse or contrapositive we need a symbol for negation. Statement Symbol Negation Symbol <3 measures 90 p <3 does not measure 90 p <3 is not acute q <3 is acute q Conditional p → q If <3 measures 90, then <3 is not acute. Converse q → p If <3 is not acute, then <3 measures 90. Inverse p → q If <3 does not measure 90, then < 3 is acute. Contrapositive q → p If <3 is acute, then <3 does not measure 90. Conditional is true, Converse is false. Inverse is false, Contrapositive is true. 4 Conditional Statement ­ p → q If the car will start, then the battery is charged. Converse ­ q → p If the battery is charged, then the car will start. Inverse ­ p → q If the car will not start, then the battery is not charged. Contrapositive ­ q → p If the battery is not charged, then the car will not start. 5 6 Deductive Reasoning uses facts, definitions and accepted properties in a logical order to write a logical argument. Inductive Reasoning uses prior experience and patterns to form a conjecture. EX) Andrea know that Sue is a sophomore and Todd is a junior. All the other juniors that Andrea knows are older than Robin. Therefore, Andrea reasons that Todd is older than Sue. What type of reasoning used? Andrea knows that Todd is older than Stan. She also knows that Stan is older than Robin. Andrea reasons that Todd is older than Robin based on the accepted statements. What type of reasoning is used? Two Laws of Deductive Reasoning ­ Law of Detachment and Law of Syllogism Law of Detachment If p → q is a true conditional statement and p is true, then q must also be true. ex) If a vehicle is a car, then it has 4 wheels. A sedan is a car. What can you conclude? p ­ a vehicle is a car q ­ it has 4 wheels p → q If a vehicle is a car, then it has 4 wheels. p A sedan is a car. (Basically just repeated the hypothesis.) Conclusion ­ q The sedan has 4 wheels. (Basically just repeated conclusion.) ex) p → q If I miss the practice the day before a game, then I will not be a starting player. I missed practice on Wednesday. p Conclusion ­ q: I will not be starting on Thursday. ex) p → q If you want a steak that's grilled to perfection, then go to Mortons. Sue went to Mortons. q Conclusion ­ No Conclusion can be made. 7 Law of Syllogism If p → q and q → r are true conditional statements, then p → r is true. ex) p → q If a metal is liquid at room temperature, then it is mercury. q → r If a metal is mercury, then its chemical symbol is Hg. p → r If a metal is liquid at room temperature, then its chemical symbol is Hg. ex) If Julie takes the car to the store, she will stop at the post office. If Julie stops at the post office, she will buy stamps. If Julie takes the car to the store, she will buy stamps. Write some conditional statements that can be made from the following statements using the Law of Syllogism. 1) If a bird is the fastest bird on land, then it is the largest of all birds. 2) If a bird is a hummingbird, then it is the smallest of all birds. 3) If a bird is the largest of all birds, then it is an ostrich. 4) If a bird is the largest of all birds, then it is flightless. 5) If a bird is the smallest bird, then it has a nest the size of a walnut half­shell. 8.