Proving Subset Relation: A⊂B ⇒ B'⊂A

[neelakash]

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.

 

[bomba923]

Consider the contrapositive:
$$A \subseteq B \to \left( {x \in A \to x \in B} \right) \to \left( {x \notin B \to x \notin A} \right) \to B' \subseteq A'$$

 

[HallsofIvy]

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?

 

[neelakash]

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 argument.

Thanks to both of you.