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
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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!
 
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  • #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.
 

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

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.

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