This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 326 Fall 2010 Homework Set 1 Solutions 2.14 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 } 2.15 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 . 2.17 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 } 2.112 (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 2.116 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....
View
Full
Document
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.
 Spring '08
 WOUTERS
 Math, Algebra

Click to edit the document details