• Support PF! Buy your school textbooks, materials and every day products Here!

Elementary proof check/help

  • Thread starter lus1450
  • Start date
  • #1
40
1

Homework Statement


For all sets A, B, and C, prove or provide a counterexample the following statements.
(A [itex]\setminus[/itex] B) [itex]\cap[/itex] (C [itex]\setminus[/itex] B) = A [itex]\setminus[/itex] (B [itex]\cup[/itex] C).


Homework Equations


N/A


The Attempt at a Solution


I went ahead and said it was false, and provided a counter example. I'm new to this and just wanna make sure my thought process was correct and the statement is indeed false.
Counterexample:
Let A = {1,2,3}, B = {2,3}, and C = {1,2}.
A [itex]\setminus[/itex] B = {1} and C [itex]\setminus[/itex] B = {1}. Then (A [itex]\setminus[/itex] B) [itex]\cap[/itex] (C [itex]\setminus[/itex] B) = {1}.
B [itex]\cup[/itex] C = {1,2,3}. Then A [itex]\setminus[/itex] (B [itex]\cup[/itex] C) = {0}. Since {1} ≠ {0}, the statement is false.

It's more the last part, i.e. "Then A [itex]\setminus[/itex] (B [itex]\cup[/itex] C) = {0}" I want to make sure is correct. Thanks for your help!
 

Answers and Replies

  • #2
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,394
179

Homework Statement


For all sets A, B, and C, prove or provide a counterexample the following statements.
(A [itex]\setminus[/itex] B) [itex]\cap[/itex] (C [itex]\setminus[/itex] B) = A [itex]\setminus[/itex] (B [itex]\cup[/itex] C).


Homework Equations


N/A


The Attempt at a Solution


I went ahead and said it was false, and provided a counter example. I'm new to this and just wanna make sure my thought process was correct and the statement is indeed false.
Counterexample:
Let A = {1,2,3}, B = {2,3}, and C = {1,2}.
A [itex]\setminus[/itex] B = {1} and C [itex]\setminus[/itex] B = {1}. Then (A [itex]\setminus[/itex] B) [itex]\cap[/itex] (C [itex]\setminus[/itex] B) = {1}.
B [itex]\cup[/itex] C = {1,2,3}. Then A [itex]\setminus[/itex] (B [itex]\cup[/itex] C) = {0}. Since {1} ≠ {0}, the statement is false.
This is almost right. Your notation {0} is incorrect. That refers to a set containing one element, the number 0. What you want is [itex]\emptyset[/itex], the empty set.

Aside from that your counterexample looks fine.
 
  • #3
40
1
I meant the empty set, but I guess 0 would be element and not empty. Thanks for telling the difference, I won't make that same mistake again.
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,772
911
If you want to use "regular" set notation, rather than ∅, it would be "{}", not "{0}".
 

Related Threads for: Elementary proof check/help

Replies
18
Views
2K
Replies
1
Views
6K
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
4
Views
878
Replies
4
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
8
Views
1K
Replies
2
Views
1K
Top