Task 1. Sets and Counterexamples 1. For each of the following statements about sets A, B and C, give a counterexample to show that the statement is false. (a) If A E B and B C C, then ACC. (b) If A E B and B E C, then A E C. Task 2. Set Operations 1. Consider the symmetric difference of the sets A and B, which is defined as: AAB := (AUB) - (ANB) to answer the following questions. (a) Using a Venn Diagram, represent AAB. (b) Is the operation A commutative? Why or why not? (c) Give a sufficient and necessary condition for AAB = AUB. 2. For every iЄ N, A₁ = (-1, ½), where A; denotes the open interval from - to. =: (a) Determine U1 A₁ = A₁ UA₂ U A3 and 1 A₁ = A₁ NA₂ NA3. Provide an explanation for your answer. (b) Determine U₁₁₁ and Ai. Provide an explanation for your answer. Note: ∞ represents infinity. Task 3. Proofs 1. Prove or disprove (AB) = AUB. 2. Prove the following statements directly from the definitions of the terms. Do not use any other facts previously proved in class or in the textbooks or in the exercises. (a) Vn Є Z, n² + n is even. (b) Va, b, c Z, if a|b and a/c, then a|(56+ 3c).
Task 1. Sets and Counterexamples 1. For each of the following statements about sets A, B and C, give a counterexample to show that the statement is false. (a) If A E B and B C C, then ACC. (b) If A E B and B E C, then A E C. Task 2. Set Operations 1. Consider the symmetric difference of the sets A and B, which is defined as: AAB := (AUB) - (ANB) to answer the following questions. (a) Using a Venn Diagram, represent AAB. (b) Is the operation A commutative? Why or why not? (c) Give a sufficient and necessary condition for AAB = AUB. 2. For every iЄ N, A₁ = (-1, ½), where A; denotes the open interval from - to. =: (a) Determine U1 A₁ = A₁ UA₂ U A3 and 1 A₁ = A₁ NA₂ NA3. Provide an explanation for your answer. (b) Determine U₁₁₁ and Ai. Provide an explanation for your answer. Note: ∞ represents infinity. Task 3. Proofs 1. Prove or disprove (AB) = AUB. 2. Prove the following statements directly from the definitions of the terms. Do not use any other facts previously proved in class or in the textbooks or in the exercises. (a) Vn Є Z, n² + n is even. (b) Va, b, c Z, if a|b and a/c, then a|(56+ 3c).
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter1: Introduction To Algebra
Section1.4: Translating Words Into Symbols
Problem 6MRE
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Transcribed Image Text:Task 1. Sets and Counterexamples
1. For each of the following statements about sets A, B and C, give a counterexample to
show that the statement is false.
(a) If A E B and B C C, then ACC.
(b) If A E B and B E C, then A E C.
Task 2. Set Operations
1. Consider the symmetric difference of the sets A and B, which is defined as:
AAB := (AUB) - (ANB) to answer the following questions.
(a) Using a Venn Diagram, represent AAB.
(b) Is the operation A commutative? Why or why not?
(c) Give a sufficient and necessary condition for AAB = AUB.
2. For every iЄ N, A₁ = (-1, ½), where A; denotes the open interval from - to.
=:
(a) Determine U1 A₁ = A₁ UA₂ U A3 and 1 A₁ = A₁ NA₂ NA3.
Provide an explanation for your answer.
(b) Determine U₁₁₁ and Ai.
Provide an explanation for your answer. Note: ∞ represents infinity.
Task 3. Proofs
1. Prove or disprove (AB) = AUB.
2. Prove the following statements directly from the definitions of the terms. Do not use any
other facts previously proved in class or in the textbooks or in the exercises.
(a) Vn Є Z, n² + n is even.
(b) Va, b, c Z, if a|b and a/c, then a|(56+ 3c).
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