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Explorations and Activities 7. Closure Explorations. In Section 1.1, we studied some of the closure prop- erties of the standard number systems. (See page 10.) We can extend this idea to other sets of numbers. So we say that: BY NO SA • A set A of numbers is closed under addition provided that whenever x and y are are in the set A, x + y is in the set A. 2.4. Quantifiers and Negations A set A of numbers is closed under multiplication provided that when- ever x and y are are in the set A, x - y is in the set A. • A set A of numbers is closed under subtraction provided that when- ever x and y are are in the set A, x - y is in the set A. For each of the following sets, make a conjecture about whether or not it is closed under addition and whether or not it is closed under multiplication. In some cases, you may be able to find a counterexample that will prove the set is not closed under one of these operations. (a) The set of all odd natural num- bers 63 (b) The set of all even integers (c) A = {1, 4, 7, 10, 13,...} (d) B = {..., -6, -3, 0, 3, 6, 9,...} (e) C = {3n+ 1\n € Z} = { ==|₁€N} (f) D =

Explorations and Activities 7. Closure Explorations. In Section 1.1, we studied some of the closure prop- erties of the standard number systems. (See page 10.) We can extend this idea to other sets of numbers. So we say that: BY NO SA • A set A of numbers is closed under addition provided that whenever x and y are are in the set A, x + y is in the set A. 2.4. Quantifiers and Negations A set A of numbers is closed under multiplication provided that when- ever x and y are are in the set A, x - y is in the set A. • A set A of numbers is closed under subtraction provided that when- ever x and y are are in the set A, x - y is in the set A. For each of the following sets, make a conjecture about whether or not it is closed under addition and whether or not it is closed under multiplication. In some cases, you may be able to find a counterexample that will prove the set is not closed under one of these operations. (a) The set of all odd natural num- bers 63 (b) The set of all even integers (c) A = {1, 4, 7, 10, 13,...} (d) B = {..., -6, -3, 0, 3, 6, 9,...} (e) C = {3n+ 1\n € Z} = { ==|₁€N} (f) D =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Need help with problems 7a and 7f

Explorations and Activities 7. Closure Explorations. In Section 1.1, we studied some of the closure prop- erties of the standard number systems. (See page 10.) We can extend this idea to other sets of numbers. So we say that: BY NO SA • A set A of numbers is closed under addition provided that whenever x and y are are in the set A, x + y is in the set A. 2.4. Quantifiers and Negations A set A of numbers is closed under multiplication provided that when- ever x and y are are in the set A, x - y is in the set A. 63 • A set A of numbers is closed under subtraction provided that when- ever x and y are are in the set A, x - y is in the set A. For each of the following sets, make a conjecture about whether or not it is closed under addition and whether or not it is closed under multiplication. In some cases, you may be able to find a counterexample that will prove the set is not closed under one of these operations. (a) The set of all odd natural num- bers (b) The set of all even integers (c) A = {1, 4, 7, 10, 13,...} (d) B = {..., -6, -3, 0, 3, 6, 9, ...} (e) C = {3n+ 1\n € Z} (f) D = = {2₁€N}
Transcribed Image Text:Explorations and Activities 7. Closure Explorations. In Section 1.1, we studied some of the closure prop- erties of the standard number systems. (See page 10.) We can extend this idea to other sets of numbers. So we say that: BY NO SA • A set A of numbers is closed under addition provided that whenever x and y are are in the set A, x + y is in the set A. 2.4. Quantifiers and Negations A set A of numbers is closed under multiplication provided that when- ever x and y are are in the set A, x - y is in the set A. 63 • A set A of numbers is closed under subtraction provided that when- ever x and y are are in the set A, x - y is in the set A. For each of the following sets, make a conjecture about whether or not it is closed under addition and whether or not it is closed under multiplication. In some cases, you may be able to find a counterexample that will prove the set is not closed under one of these operations. (a) The set of all odd natural num- bers (b) The set of all even integers (c) A = {1, 4, 7, 10, 13,...} (d) B = {..., -6, -3, 0, 3, 6, 9, ...} (e) C = {3n+ 1\n € Z} (f) D = = {2₁€N}
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