Proving or Disproving (A-B)'=B'-A' in (a-b) Homework | Steps & Examples

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Homework Help Overview

The discussion revolves around proving or disproving the set equality (A-B)' = B' - A', where A and B are defined as sets. The participants explore the implications of set operations and complements within a defined universe.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants attempt to understand the meaning of (A-B)' and question the definitions of A and B. They explore counterexamples and specific cases to analyze the set operations involved.

Discussion Status

Some participants have provided specific examples of sets and their complements, while others are still seeking clarity on the implications of their findings. There is an ongoing exploration of the relationships between the sets and their complements without a clear consensus on the proof or disproof of the original statement.

Contextual Notes

Participants are working within a defined universe of elements, and there are mentions of specific sets and their complements. The discussion includes attempts to clarify the definitions and implications of set operations, with some participants expressing uncertainty about their reasoning.

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Homework Statement


Prove or disprove:

(A-B)'=B'-A'


Homework Equations





The Attempt at a Solution


Let x[tex]\in[/tex](A-B)'
Then x[tex]\notin[/tex](A-B)
I'm not sure where to go from here...
 
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What's the context here? What are A and B? What does (A - B)' mean?
 


A-B is the set of elements in A that are not in B
So x is not in A-B means that x is in A but is not in B

You may want to try to think of some counterexamples before trying to show inclusion both ways.
 


VeeEight said:
A-B is the set of elements in A that are not in B
So x is not in A-B means that x is in A but is not in B

You may want to try to think of some counterexamples before trying to show inclusion both ways.

Ok so:
x[tex]\in[/tex]A and x[tex]\notin[/tex]B

Ok, so suppose A={1,2,3,4,5} B={3,4,6}
Then A-B={1,2,5}
So, (A-B)'={3,4,6}
so, (A-B)'=B
 


If you are working in R, then the complement of the set A-B would be R - {1, 2, 5}
You might want to try some simpler examples like A= (0,1) or {1, 2, 3} and B = [0,1] or {3, 4}
 


VeeEight said:
If you are working in R, then the complement of the set A-B would be R - {1, 2, 5}
You might want to try some simpler examples like A= (0,1) or {1, 2, 3} and B = [0,1] or {3, 4}

A={1,2,3}
B={3,4}
universe ={1,2,3,4,5,6,7}
A-B={1,2}
(A-B)'={3,4,5,6,7}
 


Okay.
 


kathrynag said:
A={1,2,3}
B={3,4}
universe ={1,2,3,4,5,6,7}
A-B={1,2}
(A-B)'={3,4,5,6,7}

VeeEight said:
Okay.
So, if x is not an element of A-B, then x is not an element of {1,2}
 


kathrynag said:

Homework Statement


Prove or disprove:

(A-B)'=B'-A'


Homework Equations





The Attempt at a Solution


Let x[tex]\in[/tex](A-B)'
Then x[tex]\notin[/tex](A-B)
I'm not sure where to go from here...
Ok, so

kathrynag said:
A={1,2,3}
B={3,4}
universe ={1,2,3,4,5,6,7}
A-B={1,2}
(A-B)'={3,4,5,6,7}

kathrynag said:
So, if x is not an element of A-B, then x is not an element of {1,2}
x[tex]\in[/tex]{3,4,5,6,7}
So x[tex]\notin[/tex]A and x[tex]\in[/tex]B
 
  • #10


kathrynag said:
A = {1,2,3} , B = {3,4} , universe = {1,2,3,4,5,6,7}

A-B = {1,2}
(A-B)' = {3,4,5,6,7}

Keep going! What are A' , B' and B'-A' ?
 
  • #11


kathrynag said:
a={1,2,3}
b={3,4}
universe ={1,2,3,4,5,6,7}
a-b={1,2}
(a-b)'={3,4,5,6,7}

pizzasky said:
keep going! What are a' , b' and b'-a' ?

a'={4,5,6,7}
b'={1,2,5,6,7}
b'-a'={1,2}
 
  • #12


Still not quite sure
Let x[tex]\in[/tex](A-B)'
x[tex]\notin[/tex](A-B).
Can I say now x[tex]\in[/tex]B? this is the part that confuses me...
 

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