Proof of Set Theory: A \subseteq B implies Bc \subseteq Ac

Also, it might be helpful to explicitly state the definition of subset.In summary, the proof states that for any sets A and B, if A is a subset of B, then B's complement is a subset of A's complement. The proof is correct and includes the definition of complement and subset.
  • #1
cmajor47
57
0

Homework Statement


For all sets A and B, if A [tex]\subseteq[/tex] B then Bc [tex]\subseteq[/tex] Ac.


Homework Equations





The Attempt at a Solution


Proof: Suppose A and B are sets and A [tex]\subseteq[/tex] B.
Let x [tex]\in[/tex] Bc
By definition of complement, if x [tex]\in[/tex] Bc then x [tex]\notin[/tex] B
Since x [tex]\notin[/tex] B, x [tex]\notin[/tex] A
Since x [tex]\notin[/tex] A, x [tex]\in[/tex] Ac by definition of complement
Therefore if A [tex]\subseteq[/tex] B then Bc [tex]\subseteq[/tex] Ac.

I just want to make sure that this proof is correct and that there are no mistakes. Thanks!
 
Physics news on Phys.org
  • #2
It's fine. You might want to enhance it's proofiness by stating the reason why x not in B implies x not in A as you gave a reason for the other lines.
 

What is the definition of "Proof of Set Theory"?

Proof of Set Theory refers to the process of using logical reasoning and mathematical principles to demonstrate the validity of statements or theorems about sets and their elements.

What does the symbol "\subseteq" mean in "A \subseteq B"?

The symbol "\subseteq" is a subset symbol and it indicates that all elements of set A are also elements of set B. In other words, set A is a subset of set B.

How does the statement "A \subseteq B implies Bc \subseteq Ac" relate to set theory?

This statement is an important theorem in set theory known as the "contrapositive" or "inverse" of the subset theorem. It states that if all elements of set A are also elements of set B, then all elements not in set B must also not be in set A.

What is the significance of proving "A \subseteq B implies Bc \subseteq Ac"?

Proving this statement is important because it helps to establish a fundamental relationship between sets and their subsets. It also allows for the use of this theorem in future mathematical proofs and computations.

How is "Proof of Set Theory" applied in real-world scientific research?

Set theory is a foundational concept in many areas of science, including computer science, physics, and biology. It is used to model and analyze complex systems, such as networks, genetic sequences, and physical phenomena. Proofs in set theory provide a rigorous and logical basis for these applications.

Similar threads

Prove every subset of countable set is either finite or else countable
    • Calculus and Beyond Homework Help
Replies
1
Views
503
Set Theory: Power sets of Unions
    • Calculus and Beyond Homework Help
Replies
3
Views
962
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
    • Calculus and Beyond Homework Help
Replies
1
Views
574
Simple set theory question
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Why is D4 not primitive on the vertices of a square?
    • Calculus and Beyond Homework Help
Replies
2
Views
930
Showing that the empty set is subset of every set
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Every subset of a finite set is finite
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Prove ##(a+b) + c = a + (b+c)## using Peano postulates
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
    • Calculus and Beyond Homework Help
Replies
1
Views
512
Topology by Simmons Problem 1.2.1
    • Calculus and Beyond Homework Help
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top