How Do You Calculate the Difference in Cardinalities of Sets A and B?

[V0ODO0CH1LD]

Scalar multiplying a set??

Homework Statement

Let A and B be two finite non-empty sets such that A $\subset$ B and n({C : C $\subset$ B\A}) = 128. Then what is the value of n(B) - n(A)?

Homework Equations

The Attempt at a Solution

I actually got to 7 by assuming that n was multiplying the cardinality of C, but I am not sure if that is what happens. What does it mean to have a scalar multiplying a set? Or is n not a scalar in this case?

 

[Mark44]

Homework Statement

Let A and B be two finite non-empty sets such that A $\subset$ B and n({C : C $\subset$ B\A}) = 128. Then what is the value of n(B) - n(A)?

Homework Equations

The Attempt at a Solution

I actually got to 7 by assuming that n was multiplying the cardinality of C, but I am not sure if that is what happens. What does it mean to have a scalar multiplying a set? Or is n not a scalar in this case?

I don't read this as "n times a set" but as "the number of elements in set <whatever>". Check your book or notes for exactly what this notation means.

 

[V0ODO0CH1LD]

I don't read this as "n times a set" but as "the number of elements in set <whatever>". Check your book or notes for exactly what this notation means.

That actually makes a lot of sense! I checked my book and n(A) is a notation for the cardinality of A. But the funny thing is that the answer would still be 7, even though I carried the notation around as if it were a multiplication.

If C = P(B\A) where P(B/A) is the power set of B\A. Then n(C : {C $\subset$ A\B}) = P(B\A) = 2^n(B\A) = 128 = 2⁷.

Therefore n(B\A) = 7.

n(B\A) = n(B) - n(A) if A $\subset$ B.

Is that still correct?

 

[Dick]

That actually makes a lot of sense! I checked my book and n(A) is a notation for the cardinality of A. But the funny thing is that the answer would still be 7, even though I carried the notation around as if it were a multiplication.

If C = P(B\A) where P(B/A) is the power set of B\A. Then n(C : {C $\subset$ A\B}) = P(B\A) = 2^n(B\A) = 128 = 2⁷.

Therefore n(B\A) = 7.

n(B\A) = n(B) - n(A) if A $\subset$ B.

Is that still correct?

Yes, it is. I'm not sure how you got it by misunderstanding the notation, but ok.