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Discrete Structures Chapter 3: The Logic of Quantified Statements 3.2 Predicates and Quantified Statements II TOUCHSTONE: Stand you both forth now: stroke your chins, and swear by your beards that I am a knave. CELIA: By our beards – if we had them – thou art. TOUCHSTONE: By my knavery – if I had it – then I were; but if you swear by that that is not, you are not forsworn. – William Shakespeare, 1564 – 1616 As You Like It, 1600 3.2 Predicates and Quantified Statements II 1 Theorem • Negation of a Universal Statement – 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 s.t. Q(x). 3.2 Predicates and Quantified Statements II 2 Example • Consider the statement: “All scientists wear glasses.” Possible Negation: “No scientist wear glasses.” Correct Negation: “There is at least one scientist who does not wear glasses.” 3.2 Predicates and Quantified Statements II 3 And this means… 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”). 3.2 Predicates and Quantified Statements II 4 Theorem • Negation of an Existential Statement – 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, s.t. Q(x)) x D, Q(x). 3.2 Predicates and Quantified Statements II 5 Example • Consider the statement: “Some snowflakes are the same.” Negation: “No snowflakes are the same.” “All snowflakes are different.” 3.2 Predicates and Quantified Statements II 6 And this means… The negation of an existential statement (“some are”) is logically equivalent to a universal statement (“none are” or “all are not”). 3.2 Predicates and Quantified Statements II 7 Example – pg. 116 # 2 • Which of the following is a negation for “All dogs are loyal”? More that one answer may be correct. a. b. c. d. e. f. g. h. All dogs are disloyal. No dogs are loyal. Some dogs are disloyal. Some dogs are loyal. There is a disloyal animal that is not a dog. There is a dog that is disloyal. No animals that are not dogs are loyal. Some animals that are not dogs are loyal. 3.2 Predicates and Quantified Statements II 8 Example – pg. 116 #3 • Write a formal negation for each of the following statements: – fish x, x has gills. – computers c, c has a CPU. – a movie m such that m is over 6 hours long. – a band b such that b has won at least 10 Grammy awards. 3.2 Predicates and Quantified Statements II 9 Negation of a Universal Conditional Statement • ~ x, if P x then Q x x such that P x and ~ Q x . • Written more symbolically ~ x, P x Q x x s.t. P x ~ Q x . 3.2 Predicates and Quantified Statements II 10 Example – pg. 116 # 17 • Write a negation for the statement below. – integers d, if 6/d is an integer then d = 3. 3.2 Predicates and Quantified Statements II 11 Variants of Universal Conditional Statements • Remember (2.2) that a conditional statement has a contrapositive, a converse, and an inverse. The definitions can be extended to universal conditional statements as shown on the next page. 3.2 Predicates and Quantified Statements II 12 Definition • Consider a statement of the form: x D, if P( x) then Q( x). 1. Its contrapositive is the statement: x D, if ~Q( x) then ~P( x). 2. Its converse is the statement: x D, if Q( x) then P( x). 3. Its inverse is the statement: x D, if ~P( x) then ~Q( x). 3.2 Predicates and Quantified Statements II 13 Example – pg. 117 # 27 • For each statement write the converse, inverse, and contrapositive. Indicate as best as you can which among the statement, its converse, its inverse, and its contrapositive are true and which are false. Give a counterexample for each that is false. – integers d, if 6/d is an integer then d = 3. 3.2 Predicates and Quantified Statements II 14 Necessary and Sufficient Conditions, Only If • Definition – The definitions of necessary, sufficient, and only if can also be extended to apply to universal conditional statements. 1. "x, r x is a sufficient condition for s x " means "x, if r x then s x ". 2. " x, r x is a neccesary condition for s x " means "x, if ~r x then ~s x " or equivalently, "x, if s x then r x " 3. " x, r x only if s x " means "x, if ~s x then ~r x " or, equivalently, "x, if r x then s x ". 3.2 Predicates and Quantified Statements II 15 Example – pg. 117 #44 & 46 • Use the facts that the negation of a statement is a statement and that the negation of an if-then statement is an and statement to rewrite the statements without using the word necessary or sufficient. – Having a large income is not a necessary condition for a person to be happy. – Being a polynomial is not a sufficient condition for a function to have a real root. 3.2 Predicates and Quantified Statements II 16