Thread: Set Theory

Cn

Originally Posted by chillerbros17

Consider sets A and B with |A| = r and |B| = s with r>s>0. Explain.

(a) What is the minimum value of |A U B|?

The minimum value is when all the elements repeat.
That is, B is contained fully in A.
Thus, A U B = A.
Thus, |A U B|=|A|=r.

(b) What is the maximum value of |A U B|?

When all elements are distincy, that is A and B are disjoint.
Then |A U B| = |A|+|B| - |A (intersect) B| =|A|+|B| *


*)By inclusion-exclusion.

(c) What is the minimum value of |A intersect B|?

When A and B are disjoint.
That is A intersect B = empty.
Thus, |A intersect B| = 0.

(d) What is the maximum value of |A intersect B|?

When B is fully contained in A.
Then, A intersect B = B
Thus,
|A intersect B|=s.