Steinhardt School of Culture, Education, and Human Development Department of Teaching and Learning. Mathematical Proof and Proving (MPP)

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1 Steinhardt School of Culture, Education, and Human Development Department of Teaching and Learning Terminology, Notations, Definitions, & Principles: Mathematical Proof and Proving (MPP) 1. A statement is a declarative sentence or assertion that is true or false (but not both). When a statement contains a variable, it should state the domain to which the variable belongs. Examples: is an even number is a statement that is True; There exists a rational number x for which x 3 0 is a statement that is True; it is a statement that contains a variable. its domain is the set of rational numbers; For all rational numbers x, x 3 0 is a statement that is False; it is a statement that contains a variable. its domain is the set of rational numbers; For all x, y R, x y 1 is a statement that is False; it is a statement that contains two variables; Their domain is the set of real numbers. All squares are rectangles is a statement that is True. The domain of the variable is the set of all squares. On January 1 st 01, at 4pm, it snowed in Washington Square Park, New York City is a statement, although we may not know at the moment whether it is true or not. Non-Examples: x 3 0 is NOT a statement; It does not state the domain of the variable; All squares are small is NOT a statement; small is not well defined; Fred owns a dog is NOT a statement. It is too vague doesn t say when. Give two examples of a statement, one with at least one variable and one without variables. Give two non-examples of a statement. Give a justification for your answers.. The negation of a statement P is the statement not P and is denoted by P. If P is True, then P is False. If P is False, then P is True. Examples: The negation of the statement is an even number is is an odd number ; The negation of the statement There exists a rational number x for which x 3 0 is the statement There does NOT exist a rational number x for which x 3 0. The negation of the statement All squares are rectangles is the statement Not all squares are rectangles, or There exists a square that is not a rectangle.

2 Write the negation of the following statements: The diagonals of a rectangle have the same length ; x R, f x 3 ; There is a smallest positive real number ; There is no real number x, for which x 1 has a solution ; If x 3, then y 5 ; 3. The disjunction of the statements P and Q is the statement P or Q ; It is denoted by P Q. The disjunction is true if at least one of P or Q is true; otherwise, P Q is false. The conjunction of the statements P and Q is the statement P and Q ; It is denoted by P Q. The conjunction is true only when both P and Q are true; otherwise, P Q is false. Example: Given that: P: A bird has two colors, one of them is blue Q: A bird has two colors, one of them is yellow Thus: P Q: A bird has two colors, one of them is yellow or blue Is the statement P Q true for a bird which is blue and yellow? Example: Given that: P: x N, x is divisible by 7 Q: x N, x is divisible by 3 Is the statement P Q true for 1? Is the statement P Q true for 3? Is the statement P Q true for 6? Is the statement P Q true for 7? 4. The implication of the statements P and Q is the statement if P, then Q ; It is denoted by P Q. The implication P Q is false when P is true and Q is false; otherwise, P Q is true. The implication P Q is often stated also as P implies Q, or Q if P, or P only if Q, or P is sufficient for Q, or Q is necessary for P. The premise of the implication P Q is P and its conclusion is Q.

3 3 We know that not all cats scratch, and we know that there are dogs that bite. What can you say about the implication If all cats scratch then there are dogs that bite? Is it true? Is the implication: If x 1, then x 1 true? What is the negation of: If x 1, then x 1? What is the negation of: If x 3, then y 5? 5. For statements P and Q, the contrapositive of the implication P Q is the implication ( Q) ( P). What is the contrapositive of the following implications? How is the implication related to its contrapositive? If x 1, then x 1 If x 3, then y 5 If food is good then it is not cheap. If a triangle has two equal angles then it is an isosceles triangle. 6. For statements P and Q, the implication Q P is called the converse of P Q. [There are places where the implication ( P) ( Q) is called the inverse of P Q. However, since it is equivalent to the converse, we will refer to both as the converse]. What is the converse of the following implications? If x 1, then x 1 If x 3, then y 5 If food is good then it is not cheap. If a triangle has two equal angles then it is an isosceles triangle. 7. Principles of Logical Inference: (a) Modus Ponens: If we know that the implication P Q is true and that P is true, then we can infer that Q must be true. (b) Modus Tollens: If we know that the implication P Q is true and that Q is true, then we can infer that P must be true. (c) Transitivity: If we know that the implication P Q is true and that the implication Q R is true, then we can infer that the implication P R must be true.

4 4 8. For statements (or open statements) P and Q, the conjunction (P Q) (Q P) of the implication P Q and its converse is called the biconditional of P and Q and is denoted by P Q. The biconditional P Q is often stated as: P is equivalent to Q, or P if and only if Q, or P is a necessary and sufficient condition for Q. 9. A compound statement is a statement composed of one or more given statements (called component statements in this context) and at least one logical connective (i.e.,,,,, ). 10. A compound statement S is called a tautology if it is true for all possible combinations of truth-values of the component statements that comprise S. 11. A compound statement S is called a contradiction if it is false for all possible combinations of truth-values of the component statements that comprise S. 1. Two compound statements S 1 and S, with the same component statements, are called logically equivalent if they have the same truth-value for every combination of truth-values of their component statements. The notation S 1 S means that S 1 and S are logically equivalent. 13. Let P(x) be an open sentence over domain D. Adding the phrase "for every x D" or the phrase "there exists x D such that" to P(x) produces a statement called a quantified statement. These phrases are called quantifiers. 14. Let P(x) be an open sentence over domain D. The quantifier "for every x D" is referred to as the universal quantifier and is denoted by the symbol. The quantified statement x D, P(x) is called a universal statement meaning that for every x in the domain D the statement P(x) is true. The quantifier "there exists x D such that" is referred to as the existential quantifier and is denoted by the symbol. The quantified statement x D, P(x) is called an existential statement meaning that there exists an x in the domain D for which the statement P(x) is true.

5 5 15. The Principle of Mathematical Induction: For any open sentence P(n), if: (i) (ii) P(1) is true the implication P(k) P(k+1) is true for every positive integer k (i.e., k ) then P(n) is true for every positive integer n (i.e., n ). A proof by Mathematical Induction has three main components: The Base Step: The step in which we verify that P(1) is true, that is, that the 'property' P holds for n=1. The Inductive Hypothesis: We assume that P(n) is true for n=k, where k is an arbitrary positive integer. The Inductive Step: The step in which, based on the Inductive Hypothesis, we prove that it follows that P(n) is true for n=k An equivalent version of the Principle of Mathematical Induction: For any open sentence P(n), if: (i) P(1) is true (ii) for every positive integer k (i.e., k ), if P(t) is true for 1 t k, then P(k+1) is true then P(n) is true for every positive integer n ( n ).

6 6 17. Methods for proving/disproving an implication P(x) Q(x): P(x) Q(x) Assume Show Direct Proof P(x) Q(x) Indirect Proof by Contradiction Indirect Proof by Contrapositive Disprove P(x) and ~Q(x) ~Q(x) A contradiction of the form: T(x) and ~T(x) (T(x) can be any statement) ~P(x) A Counter-Example: Find an x for which P(x) and ~Q(x) 18. Methods for proving/disproving existential statements x D, P(x): x D, P(x) Assume Show Prove by construction Find or construct an x in D for which P(x) Prove by inference Prove/show that there must be an x in D for which P(x) Disprove with an Indirect Proof by Contradiction x D, P(x) A contradiction of the form: T(x) and ~T(x) (T(x) can be any statement)