Consider the following statement. For all sets A and B, (A ∪ Bc) − B = (A − B) ∪ Bc. An algebraic proof for the statement should cite a property from Theorem 6.2.2 for every step, but some reasons are missing from the proposed proof below. Indicate which reasons are missing. (Select all that apply.) Let any sets A and B be given. Then (A ∪ Bc) − B = (A ∪ Bc) ∩ Bc by the set difference law (1) = (Bc ∩ A) ∪ (Bc ∩ Bc) by the distributive law (2) = (Bc ∩ A) ∪ Bc by the idempotent law for ∪ (3) = (A − B) ∪ Bc by the set difference law (4) The absorption law is needed between steps (2) and (3).The double complement law is needed between steps (3) and (4).The complement law is needed between steps (2) and (3).The commutative law is needed between between steps (1) and (2).The commutative law is needed between between steps (3) and (4).
Consider the following statement. For all sets A and B, (A ∪ Bc) − B = (A − B) ∪ Bc. An algebraic proof for the statement should cite a property from Theorem 6.2.2 for every step, but some reasons are missing from the proposed proof below. Indicate which reasons are missing. (Select all that apply.) Let any sets A and B be given. Then (A ∪ Bc) − B = (A ∪ Bc) ∩ Bc by the set difference law (1) = (Bc ∩ A) ∪ (Bc ∩ Bc) by the distributive law (2) = (Bc ∩ A) ∪ Bc by the idempotent law for ∪ (3) = (A − B) ∪ Bc by the set difference law (4) The absorption law is needed between steps (2) and (3).The double complement law is needed between steps (3) and (4).The complement law is needed between steps (2) and (3).The commutative law is needed between between steps (1) and (2).The commutative law is needed between between steps (3) and (4).
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.3: Divisibility
Problem 8TFE
Related questions
Question
Consider the following statement.
For all sets A and B,
(A ∪ Bc) − B = (A − B) ∪ Bc.
An algebraic proof for the statement should cite a property from Theorem 6.2.2 for every step, but some reasons are missing from the proposed proof below. Indicate which reasons are missing. (Select all that apply.)
Let any sets A and B be given. Then
| (A ∪ Bc) − B | = | (A ∪ Bc) ∩ Bc | by the set difference law | (1) | |
| = | (Bc ∩ A) ∪ (Bc ∩ Bc) | by the distributive law | (2) | ||
| = | (Bc ∩ A) ∪ Bc | by the idempotent law for ∪ | (3) | ||
| = | (A − B) ∪ Bc | by the set difference law | (4) |
The absorption law is needed between steps (2) and (3).The double complement law is needed between steps (3) and (4).The complement law is needed between steps (2) and (3).The commutative law is needed between between steps (1) and (2).The commutative law is needed between between steps (3) and (4).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,