set theory

[Var1995]

2nd statement is wrong right as both the sets are non-empty so their union can't be a null set

[Obi Wan]

Only statement 3 is correct. Hence answer is option (b)

[Dr. Strange]

The second statement is also correct because for a union to be a null set both A-B and B-A should be  null sets .i.e A=B.

So A-B= NULL
     B-A =NULL  
     Their union is NULL
     Thus A+B is NULL.

[Var1995]

but according to the question 2nd statement can't be right as the sets are non- empty

[Dr. Strange]

A+B here is union between A-B and B-A
So B-A and A-B need to be null sets and not A and B
A and B are non empty but A-B and B-A are both null sets.

[Amit Goyal]

I have posted the answer here:

http://economicsentrance.in/thread/sets/

[Dr. Strange]

Statement 1 should be incorrect as shown in the pic

[Amit Goyal]

Hi,

You have proved that the converse of statement 1 is false, I agree with that. But statement 1 is true, for the proof refer the solution I have posted.

[Dr. Strange]

I have a doubt.

Since I have proven this statement to be true (which applies in this case as A,B are not null sets) :

If A is a subset of B, A+B is not equal to B

So its contrapositive should be true which is

If A+B is equal  to B , A is not a subset of B

Don't you think it is contradicting the statement.

[Amit Goyal]

You have proven that the converse is FALSE.

This is our original statement, let us call it statement X:  If A + B = B then A is a subset of B.
This is its contrapositive, let us call it statement C(X): If A is not a subset of B the A+B and B are not equal.

I have shown that C(X) is true in the link and since X and C(X) are equivalent we have proved X.

This is the converse of X, let us call it X': If A is a subset of B then A+B = B.
You have shown that X' is false, i.e. you have shown that the following statement is true, let us call it N(X'): It is possible in some case that A is a subset of B and A+B and B are not equal.
The above statement N(X') is not equivalent to: If A+B and B are equal then  A is not a subset of B.
Infact, N(X') (that you have shown to be true) is not an implication and hence does not have a contrapositive.

For logic, I recommend http://arrogant.stanford.edu/intrologic/

[Dr. Strange]

It is possible in some case that A is a subset of B and A+B and B are not equal.
I agree with the statement that it is not an implication.
However if we can prove that  IF A is a subset of B THEN A+B and B are not equal - it is an implication and we can take its contrapositive to be true.

Proof:

Since A is subset of B.
So A-B =NULL
So A-B UNION B-A becomes B-A
However A is not null and a subset of B.
So B-A can NEVER be equal to B.
Hence we have proved the implication that if A is a subset of B THEN A+B and B are not equal.
Hence its contrapositive should be true.

[Amit Goyal]

Both these statements are true at the same time:

Statement 1: for all A, B: (A + B) = B implies A ⊂ B  (I proved)
Statement 2: for all A ≠ ∅, B ≠ ∅: (A + B) = B implies A ⊄ B  (You proved)

Since statement 1 is true, this is also true:
Statement 3: for all A ≠ ∅, B ≠ ∅: (A + B) = B implies A ⊂ B (Statement 1 implies Statement 3)

The reason why all the statements 1, 2 and 3 are true is (A + B) = B implies A = ∅. Therefore, statement 1 is true because empty set is the subset of every set and statement 2 and 3 are true because (A + B) and B are never equal for non-empty sets A, B.

[Dr. Strange]

Perfectly explained.
Thanks a lot.

[Var1995]

i m not getting the 2nd implication in the proof of 2nd statement. plz help. have we used the result of statement 1 to prove it?

[Rootz]

I have a small doubt. Doesnt the first statement say A + B = B implies A Is a subset of B or equal to B? Isnt there a small line under the subset sign?