Question

True or False? If false, give a counterexample (sets A and B).

a. If A ⊆ B with B infinite, then A is infinite.
b. If A ⊆ B with A infinite, then B is infinite.
c. If A ⊆ B with B finite, then A is finite.
d. If A ⊆ B with A finite, then B is finite.

Answer 1

Answers:

1. a = false
2. b = true
3. c = true
4. d = false

Explanation:

1. The first statement is false because of the counterexample A = {1,2,3} and B = set of all real numbers. A is a subset of B, where B has infinitely items in it, but A is finite.
2. The second statement is true since B must be the same size as A or larger. Think of the ⊆ symbol as "less than or equal to", but it pertains to comparing sets rather than comparing numbers. Since A is infinitely large, so is B. An example would be A = set of rational numbers, B = set of real numbers. Both sets have infinitely many numbers in them.
3. This statement is true. With B having finitely many members, it constricts how many members A could have. Set A has the same number of values as B, or fewer values. An example: A = {1,2,3} and B = {1,2,3,4}. If sets A and B have the same number of values, then A = B.
4. This is false. Refer to statement (a) shown above. I mentioned that A = {1,2,3} and B = set of all real numbers. Feel free to explore other counterexamples.