HWSET1-SOL - Math 326 Fall 2010 Homework Set 1 Solutions...

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Unformatted text preview: Math 326 Fall 2010 Homework Set 1 Solutions 2.1-4 Let Q ( n ) be the predicate “ n 2 ≤ 30.” (a) Write Q (2), Q (- 2), Q (7), and Q (- 7) and indicate which of these statements are true and which are false. Q (2) ⇐⇒ 2 2 ≤ 30 ⇐⇒ 4 ≤ 30 ⇐⇒ T Q (- 2) ⇐⇒ (- 2) 2 ≤ 30 ⇐⇒ 4 ≤ 30 ⇐⇒ T Q (7) ⇐⇒ 7 2 ≤ 30 ⇐⇒ 49 ≤ 30 ⇐⇒ F Q (- 2) ⇐⇒ (- 7) 2 ≤ 30 ⇐⇒ 49 ≤ 30 ⇐⇒ F (b) Find the truth set of Q ( n ) if the domain of n is Z . To find the truth set we must solve n 2 ≤ 30 for all values of n that are integers, i.e.,- √ 30 ≤ n ≤ √ 30. The truth set is thus {- 5 ,- 4 ,- 3 ,- 2 ,- 1 , , 1 , 2 , 3 , 4 , 5 } (c) If the domain is the set Z + of all positive integers, what is the truth set of Q ( n )? To find the truth set we must solve n 2 ≤ 30 for all values of n that are positive integers, i.e., 0 < n ≤ √ 30. The truth set is thus { 1 , 2 , 3 , 4 , 5 } 2.1-5 Let Q ( x,y ) be the predicate “If x < y then x 2 < y 2 ” with domain for both x and y being the set R of real numbers. (a) Explain why Q ( x,y ) is false if x =- 2 and y = 1. Q (- 2 , 1) says (- 2 < 1) = ⇒ (4 < 1), which is T = ⇒ F , which is F . (b) Give values different from those in part (a) for which Q ( x,y ) is false. Try x =- 1 and y = 0. Then Q (- 1 , 0) says- 1 < 0 = ⇒ 1 < 0, which is also T = ⇒ F , and hence F . (c) Explain why Q ( x,y ) is true if x = 3 and y = 8. Q (3 , 8) says (3 < 8) = ⇒ (9 < 64) which is T = ⇒ T which is T . (d) Give values different from those in part (c) for which Q ( x,y ) is true. Try x = 2 and y =- 3. Then Q (2 ,- 3) says (2 <- 3) = ⇒ (4 < 9) which is F = ⇒ T which is T . 2.1-7 Find the truth of each predicate. 1 (a) predicate: 6 /d is an integer, domain Z . {- 6 ,- 3 ,- 2 ,- 1 , 1 , 2 , 3 , 6 } (b) predicate: 6 /d is an integer, domain Z + . { 1 , 2 , 3 , 6 } (c) predicate: 1 ≤ x 2 ≤ 4, domain R . { x ∈ R | - 2 ≤ x ≤ - 1 ∨ 1 ≤ x ≤ 2 } (d) predicate: 1 ≤ x 2 ≤ 4, domain Z . {- 2 ,- 1 , 1 , 2 } 2.1-12 (Find counterexamples to show that the statement is false): ∀ real numbers x and y , √ x + y = √ x + √ y . Try x = y = 1. Then √ x + y = √ 1 + 1 = √ 2 6 = 2 = 1 + 1 = √ 1 + √ 1 = √ x + √ y 2.1-16 Rewrite each of the following statements in the form “ ∀ x , .” (a) All dinosaurs are extinct. ∀ dinosaur x,x is extinct (b) Every real number is positive, negative, or zero. ( ∀ x ∈ R )(( x > 0) ∨ ( x < 0) ∨ ( x = 0)) (c) No irrational numbers are integers. ∀ irrational number x,x 6∈ Z (d) No logicians are lazy. ∀ logicians x,x is not lazy (e) The number 2,147,581,953 is not equal to the square of any integer. ∀ m ∈ Z ,m 2 6 = 2 , 147 , 581 , 953 (f) The number -1 is not equal to the square of any real number....
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This note was uploaded on 06/12/2011 for the course MATH 103 taught by Professor Wouters during the Spring '08 term at Wisc Oshkosh.

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HWSET1-SOL - Math 326 Fall 2010 Homework Set 1 Solutions...

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