Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CHAPTER TWO Conditional Statement a logical statement that is dependent upon some other factor. Conditional Statements are written in an “If – Then” format. First part of a Conditional Statement is called the hypothesis. This is the assumption. The hypothesis always follows directly after the “If” in the “If-Then” format. “If” is not part of the hypothesis. The hypothesis is symbolized by the italicized lower case p. Second part of a Conditional Statement is called the conclusion. This is what happens if the assumption is true. The conclusion always follows directly after the “Then” in the “If-Then” format. “Then” is not part of the conclusion. The conclusion is symbolized by the italicized lower case q. Conditional Statements can be either true or false. This is called its Truth Value. A true hypothesis does not dictate a true Conditional Statement and vice versa. To show that the Conditional Statement is false, you have to show only one counterexample. Statements that are based on a given conditional statement are called Related Conditionals and they have special names (Converse, Inverse, Contrapositive). “Then” is abbreviated with a small arrow to the right . If is not included in the symbolic representation of the Conditional or its Related Conditionals. Converse of a conditional statement is formed by switching the hypothesis and conclusion. The Converse of a conditional statement may not always be true even if the original Conditional Statement is true. Negation is the negative of a statement. It is formed by adding the word “not”. It is symbolized by a swoop in front of the p, ~ p. If a statement is true than its negation is false and vice versa. The truth values can be organized in a truth table. p ~p T F F T Inverse of a Conditional Statement is formed by negating both the hypothesis and conclusion of a conditional statement. Contrapositive of a Conditional Statement is formed by negating the Converse of the Conditional Statement. Law of Contrapositive is the relationship of the truth values of a conditional and its contrapositive. Equivalent Statements or Logically Equivalent Statements are when the two statements are both true or both false. Conditional Statement = Contrapositive Inverse = Converse IF p, THEN q Conditional Statement IF q, THEN p Converse IF not ~p, THEN not ~q Inverse IF not ~q, THEN not ~p Contrapositive Perpendicular Lines are two lines who intersect to form four right angles. If two lines intersect to form four right angles then the lines are perpendicular Line Perpendicular to a Plane is a line that intersect the plane in a point and is perpendicular to every line in the plane that intersects it. is the symbol for “perpendicular” If a Conditional Statement is true and its Converse is true then you can combine the two into a Biconditional Statement. A Biconditional Statement is a statement that contains the phrase “if and only if”. Biconditional Statements can be true or false. All definitions can be written as true biconditional statements. Unlike definitions, not all postulates can be written as true biconditional statements. “if he committed a crime, then he was at the scene of the crime” “If and Only If” can be abbreviated “iff” In a Biconditional Statement, the “if and only if” can be symbolized as a two way arrow . pq Conditional Statement qp Converse ~p ~q Inverse ~q ~p Contrapositive pq Biconditional The negation of a negative statement is a positive statement. Deductive Reasoning uses facts, definitions and accepted properties in a logical order to write a logical argument. Deductive Reasoning is using definite information to form a definitive conclusion while Inductive Reasoning uses definite information or circumstantial information to form a educated guess as to the conclusion. One method of solving problems using Deductive Reasoning is called matrix logic where a grid or table is made to solve the problem. A Conjunction is a compound statement formed by joining two statements with the word “and”. A Conjunction is symbolized by . A conjunction is true only when both the statements are true. A Disjunction is a compound statement formed by joining two statements with the word “or”. A Disjunction is symbolized by . A disjunction is false only when both statements are false. The Law of Detachment states that if the conditional statement is true then the hypothesis is true and the conclusion is true. When you apply the Law of Detachment, make sure that the conditional is true before you test the validity of the conclusion. Law of Detachment symbolically is as follows: [(p q) p] q The Law of Syllogism states That if a hypothesis leads to a conclusion and that conclusion in turn forms a hypothesis that leads to another conclusion then the original hypothesis can be said to lead to the second conclusion. If p q and q r are true conditional statements, then p r is true. Law of Syllogism symbolically is as follows: [(p q) (q r)] (p r) The original Conditional Statement must be true to use the Law of Syllogism. Addition Property If a =b, then a + c = b + c Subtraction Property If a =b, then a - c = b – c Multiplication Property If a = b, then ac = bc Division Property If a = b and c 0, then a c = b c Reflexive Property a=a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c Substitution Property If a = b, a + 5 = 7 can be written b + 5 = 7 Distributive Property a(b + c) = ab + ac Commutative Property of Addition doesn’t matter what order you add the numbers the result will be the same Commutative Property of Multiplication doesn’t matter what order you multiply the numbers the result will be the same Associative Property of Addition doesn’t matter how the numbers are grouped before you add the numbers the result will be the same Associative Property of Multiplicationdoesn’t matter how the numbers are grouped before you multiply the numbers the result will be the same Deductive Argument is a group of algebraic steps used to solve problems Theorem is a true statement that follows as a result of other true statements. Two-Column Proof is a two-column table that has numbered statements and reasons that show the logical order of an argument. Each step of a two-column proof is called a statement. In a two-column proof, the first column on the left is always the statements and labeled as such. In a two-column proof, the second column on the right is always the reasons that justify the statements and labeled as such. In a two-column proof, you cannot write a statement unless you can give a reason for it. Paragraph Proof , also called an informal proof, is a statements and reasons with a logical order of argument written as a paragraph. Before writing a proof, you should have a plan. One strategy is to work backward. Start with what you want to prove and work backward step by step until you reach the given information. 5 essential parts of a good proof: 1. State the theorem or conjecture to be proven. 2. List the given information. 3. If possible, draw a diagram to illustrate the given information. 4. State what is to be proved. 5. Develop a system of deductive reasoning. Ruler Postulate states the points on any line or line segment can be paired with real numbers like on a ruler so that given any two points A and B on a line, A corresponds to zero, and B corresponds to a positive real number. Remember AB with no line over it is the symbol for the measure of a line segment and not the name of the line segment itself. Reflexive Property a a Symmetric Property If a b, then b a Transitive Property If a b and b c, then a c Right Angle Congruence Theorem states that all right angles are congruent. Congruent Supplements Theorem states that if two angles are supplementary to the same angle (or to congruent angles) then they are congruent. If m1 + m2 = 180 and m2 + m3 = 180, then 1 3. Notice then when talking about measure you use the equal sign and when you are talking about the angles themselves you use the congruence sign. Complement Theorem states that if the non-common sides of two adjacent angles form a right angle, then the angles are complementary angles. Congruent Complements Theorem states that if two angles are complementary to the same angle (or to congruent angles) then they are congruent. Linear Pair Postulate states that if two angles form a linear pair, then they are supplementary. Vertical Angles Theorem state that all vertical angles are congruent. Protractor Postulate Given ray AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of ray AB, such that the measure of the angle formed is r. Angle Addition Postulate states that two adjacent angles can add together to make one larger angle so that the sum of the measures of each of the smaller two adjacent angles equals the angle measure of the larger angle. P Q R S If mPQR + mRQS = mPQS, then the ray QR is in the interior of mPQS