Proving Subset Relation: A⊂B ⇒ B'⊂A

In summary, the conversation discusses proving that if A is a subset of B, then B' is a subset of A'. The attempt at a solution involves considering the contrapositive and using the definition of subsets. The error in the initial argument is pointed out and the correct approach is suggested.
  • #1
neelakash
511
1

Homework Statement



I have to prove that if A blis a subset of B then B' is a subset of A'.

Homework Equations




The Attempt at a Solution



I did:
Let x belongs to B but x does not belong to A
=>x does not belong to B' but x belongs to A'
Hence proved.

please tell me if I am correct.
 
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  • #2
Consider the contrapositive:
[tex]A \subseteq B \to \left( {x \in A \to x \in B} \right) \to \left( {x \notin B \to x \notin A} \right) \to B' \subseteq A'[/tex]
 
Last edited:
  • #3
neelakash said:

Homework Statement



I have to prove that if A blis a subset of B then B' is a subset of A'.

Homework Equations




The Attempt at a Solution



I did:
Let x belongs to B but x does not belong to A
=>x does not belong to B' but x belongs to A'
Hence proved.

please tell me if I am correct.
How does "x does not belong to B' but does belong to A' " prove B' is a subset of A'?
For example, if B' were {1, 2, 3, 4, 5} and A' were {5, 6, 7} then x= 6 is not in B' but is in A'. It is certainly not the case that "B' is a subset of A'"!

To prove "B' is a subset of A'", you must, using the definition, prove "If x is in B' then it is in A'.

If x is in B', then what can you say about x?
 
  • #4
How does "x does not belong to B' but does belong to A' " prove B' is a subset of A'?
You are correct.I was wrong in that arguement.

Thanks to both of you.
 

FAQ: Proving Subset Relation: A⊂B ⇒ B'⊂A

1. What does "A⊂B" mean in the context of subset relation?

In mathematics, "A⊂B" means that set A is a subset of set B, which indicates that all elements in set A are also present in set B.

2. How is the subset relation "A⊂B" proven?

The subset relation "A⊂B" can be proven by showing that every element in set A is also present in set B. This can be done through a direct proof, a proof by contradiction, or a proof by contrapositive.

3. What is the significance of "B'⊂A" in the statement "A⊂B ⇒ B'⊂A"?

The statement "A⊂B ⇒ B'⊂A" means that if set A is a subset of set B, then the complement of set B is a subset of the complement of set A. This is significant because it shows the relationship between the original sets and their complements.

4. Can the statement "A⊂B ⇒ B'⊂A" be proven using only one set?

No, the statement "A⊂B ⇒ B'⊂A" requires the use of two sets - A and B. This is because the statement is about the subset relation between two sets and their respective complements.

5. How is the statement "A⊂B ⇒ B'⊂A" used in practical applications?

The statement "A⊂B ⇒ B'⊂A" is used in various areas of mathematics and science, such as in set theory, graph theory, and computer science. It is also used in proving theorems and solving problems related to subsets and complements of sets.

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