Question

If A and B are two finite sets. Prove that |A ∪ B| = |A| + |B| − |A ∩ B| is true.

Answer #1

3. Prove or disprove the following statement: If A and B are
finite sets, then |A ∪ B| = |A| + |B|.

Prove the union of a finite collection of countable
sets is countable.

Assume that X and Y are finite sets. Prove the following
statement:
If there is a bijection f:X→Y then|X|=|Y|.
Hint: Show that if f : X → Y is a surjection then |X| ≥ |Y| and if
f : X → Y is an injection then
|X| ≤ |Y |.

Prove that the union of two compact sets is compact using the
fact that every open cover has a finite subcover.

Prove the following for the plane.
a.) The intersection of two closed sets is closed.
b.) The intersection of two open sets is open.

.Unless otherwise noted, all sets in this module are finite.
Prove the following statements!
1. There is a bijection from the positive odd numbers to the
integers divisible by 3.
2. There is an injection f : Q→N.
3. If f : N→R is a function, then it is not surjective.

Prove for each:
a. Proposition: If A is finite and B is countable, then A ∪ B is
countable.
b. Proposition: Every subset A ⊆ N is finite or countable.
[Similarly if A ⊆ B with B countable.]
c. Proposition: If N → A is a surjection, then A is finite or
countable. [Or if countable B → A surjection.]

Prove that for all sets A, B, C,
A ∩ (B ∩ C) = (A ∩ B) ∩ C
Prove that for all sets A, B,
A \ (A \ B) = A ∩ B.

Using the following theorem: If A and B are disjoint denumerable
sets, then A ∪ B is denumerable, prove the union of a finite
pairwise disjoint family of denumerable sets {Ai
:1,2,3,....,n} is denumerable

Let A and B be sets. Prove that A ⊆ B if and only if A − B =
∅.

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