Consider the following theorem. Theorem: For all sets A and B, if A C B, then A U B CB. Supply explanations for the steps in the proof of the theorem. Proof: Statement Explanation Suppose A, B, and C are any sets such that AC B. starting point We must show that A UBC B. conclusion to be shown Let x be any element in AU B. start of an element proof Then x is in A or x is in B. ---Select--- In case x is in A, then x is in B ---Select--- In case x is in B, then x is in B. tautology (p → P). So in either case, x is in B. proof by division into cases Thus every element in A UB is in B. since x could be any element of A U B Therefore, A UBCB --Select--- [as was to be shown].
Consider the following theorem. Theorem: For all sets A and B, if A C B, then A U B CB. Supply explanations for the steps in the proof of the theorem. Proof: Statement Explanation Suppose A, B, and C are any sets such that AC B. starting point We must show that A UBC B. conclusion to be shown Let x be any element in AU B. start of an element proof Then x is in A or x is in B. ---Select--- In case x is in A, then x is in B ---Select--- In case x is in B, then x is in B. tautology (p → P). So in either case, x is in B. proof by division into cases Thus every element in A UB is in B. since x could be any element of A U B Therefore, A UBCB --Select--- [as was to be shown].
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![Consider the following theorem.
Theorem: For all sets A and B, if A C B, then A U B CB.
Supply explanations for the steps in the proof of the theorem.
Proof:
Statement
Explanation
Suppose A, B, and C are any sets such that A C B.
starting point
We must show that A UBC B.
conclusion to be shown
Let x be any element in AU B.
start of an element proof
Then x is in A or x is in B.
---Select---
In case x is in A, then x is in B
-Select---
In case x is in B, then x is in B.
tautology (p → p)
So in either case, x is in B.
proof by division into cases
Thus every element in A UB is in B.
since x could be any element of A U B
---Select---
Therefore, AUBC B
[as was to be shown].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa28e667e-d0bd-42fc-8ff7-fbe8b481d237%2F9c6286b0-16b2-42cd-91c3-356c86f14e74%2F6ychshg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the following theorem.
Theorem: For all sets A and B, if A C B, then A U B CB.
Supply explanations for the steps in the proof of the theorem.
Proof:
Statement
Explanation
Suppose A, B, and C are any sets such that A C B.
starting point
We must show that A UBC B.
conclusion to be shown
Let x be any element in AU B.
start of an element proof
Then x is in A or x is in B.
---Select---
In case x is in A, then x is in B
-Select---
In case x is in B, then x is in B.
tautology (p → p)
So in either case, x is in B.
proof by division into cases
Thus every element in A UB is in B.
since x could be any element of A U B
---Select---
Therefore, AUBC B
[as was to be shown].
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