DISCRETE MATH: LECTURE 4


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1 DISCRETE MATH: LECTURE 4 DR. DANIEL FREEMAN 1. Chapter 3.1 Predicates and Quantified Statements I A predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables. The domain of a predicate variable is the set of all values that may be substituted in place of the variable. For Example: Consider the sentences Daniel is a Fightin Texas Aggie, and Daniel is a graduate of Texas A&M University. If P (x) is a predicate and x has domain D, the truth set of P (x) is the set of all elements of D that make P (x) true when they are substituted for x. The truth set of P (x) is denoted: {x D P (x)}. For Example: Let P (x) be the predicate x 2 > x with domain R. Write P (2), P ( 1 2 ), and P ( 1 2 ), and indicate which of these statements are true and which are false The Universal Quantifier:. Quantifiers are words that refer to quantities such as some or all and tell for how many elements a given predicate is true. The symbol denotes for all and is called the universal quantifier. Let Q(x) be a predicate and D the domain of x. A universal statement is a statement of the form x D, Q(x). It is defined to be true if, and only if, Q(x) is true for every x in D. It is defined to be false if, and only if, Q(x) is false for at least one x in D. A value for x for which Q(x) is false is called a counterexample to the universal statement. For Example: We can express the sentence All human beings are mortal, as human beings x, x is mortal. 1
2 2 DR. DANIEL FREEMAN Your Example: Write down a universal statement. Define the domain and its truth set. Then, write down a false universal statement, give the domain, and give a counterexample The Existential Quantifier:. The symbol denotes there exists and is called the existential quantifier. Let Q(x) be a predicate and D the domain of x. An existential statement is a statement of the form x D such that Q(x). It is defined to be true if, and only if, Q(x) is true for at least one x in D. It is false if, and only if, Q(x) is false for all x in D. For Example: m Z + such that m 2 = m. For Example: Let E = {5, 6, 7, 8}. m E such that m 2 = m Universal Conditional Statements. A universal conditional statement has the form: x, if P (x) then Q(x). For Example: Rewrite each of the following statements in the form: then. (1) If a real number is an integer, then it is a rational number. (2) All bytes have eight bits. (3) No fire trucks are green., if
3 DISCRETE MATH: LECTURE Equivalent Forms of Universal and Existential Statements. x U, if P (x) then Q(x) is equivalent to x D, Q(x) x such that P (x) and Q(x) is equivalent to x D such that Q(x). 2. Predicates and Quantified Statements II: The Quantifiers Strike Back The negation of a statement of the form x in D, Q(x) is logically equivalent to a statement of the form x in D such that Q(x). Symbolically, ( x D, Q(x)) x D such that Q(x). The negation of a universal statement ( all are ) is logically equivalent to an existential statement ( some are not or there is at least one that is not ). When we speak of logical equivalence for quantified statements, we mean that the statements always have identical truth values no matter what predicates are substituted for the predicate symbols and not matter what sets are used for the domains of the predicate variables. The negation of a statement of the form x in D such that Q(x) is logically equivalent to a statement of the form x in D, Q(x). Symbolically, ( x D such that Q(x)) x D, Q(x). The negation of an existential statement ( some are is logically equivalent to a universal statement ( none are or all are not ) In Class Work. Determine whether the proposed negation is correct. If it is not, write a correct negation. Statement Proposed Negation 1 The sum of any two irrational numbers The sum of any two irrational numbers is rational. numbers is irrational 2 For all integers n, For all integers n, if n 2 is even then n is not even. if n 2 is even then n is even Negations of Universal Conditional Statements. The negation of a universal conditional statement is ( x, if P (x) then Q(x)) x such that P (x) and Q(x).
4 4 DR. DANIEL FREEMAN 2.3. In Class Work. Write a negation for each statement: (1) real numbers x, if x 2 1 then x > 0. (2) x R, if x(x + 1) > 0 then x > 0 or x < 1. (3) integers a, b, and c, if a b is even and b c is even, then a c is even. (4) If the square of an integer is odd, then the integer is odd The Relation Among,,, and. In a sense, universal statements are generalizations of and statements, and existential statements are generalizations of or statements. If Q(x) is a predicate and the domain D of x is the set {x 1, x 2,..., x 3 }, then the statements x D, Q(x) and Q(x 1 ) Q(x 2 ) Q(x n ) are logically equivalent. Similarly, if Q(x) is a predicate and D = {x 1, x 2,..., x n }, then the statements and x D such that Q(x) Q(x 1 ) Q(x 2 ) Q(x n ) are logically equivalent. This is why the negation of statements with the quantifiers for all and there exists are analogous to the De Morgan s Laws Vacuous Truth of Universal Statements. In general, a statement of the form x in D, if P (x) then Q(x) is called vacuously true or true by default if, and only if, P (x) is false for every x in D Your Example: Write down a universal statement that is vacuously true.
5 DISCRETE MATH: LECTURE Variants of Universal Conditional Statements. Consider a statement of the form: x D, if P (x) then Q(x). Its contrapositive is the statement x D, if Q(x) then P (x) Its converse is the statement x D, if Q(x) then P (x). Its inverse is the statement x D, if P (x) then Q(x) 2.7. In Class Work. Write the contrapositive, converse, and inverse of the following statement: If a real number is greater than 2, then its square is greater than 4. Show that a universal conditional statement is logically equivalent to its contrapositive. Show that a universal conditional statement is NOT logically equivalent to its converse. Show that a universal conditional statement is NOT logically equivalent to its inverse.
6 6 DR. DANIEL FREEMAN 2.8. Necessary and Sufficient Conditions, Only If. x, r(x) is a sufficient condition for s(x) means x, if r(x) then s(x). x, r(x) is a necessary condition for s(x) means x, if r(x) then s(x) or, equivalently, x, if s(x) then r(x). x, r(x) only if s(x) means x, if s(x) then r(x), or equivalently, x, if r(x) then s(x) In Class Work. Rewrite the following statements as quantified conditional statements. Do not use the word necessary or sufficient. (1) Squareness is a sufficient condition for rectangularity. (2) Being at least 35 years old is a necessary condition for being President of the United States. Rewrite the following as a universal conditional statement: A product of two numbers is 0 only if one of the numbers is 0.
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