discrete_mathematics_with_applications/chapter_6/test_yourself.md

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Test Yourself

1. The notation A \subseteq B is read "_____" and means that _____.

The set A is a subset of the set B; if x \in A then x \in B

2. To use an element argument for proving that a set X is a subset of a set Y, you suppose that _____ and show that _____.

x is a particular but arbitrarily chosen element of X; x is an element of Y.

3. To disprove that a set X is a subset of a set Y, you show that there is _____.

an element in X that is not in Y.

4. An element x is in A \cup B if, and only if, _____.

x is in either A or B.

5. An element x is in A \cap B if, and only if, _____.

x is in both A and B.

6. An element x is in B - A if, and only if, _____.

x is in B but not in A.

7. An element x is in A^c if, and only if, _____.

x is in the universal set and is not in A.

8. The empty set is a set with _____.

no elements.

9. The power set of a set A is _____.

the set of all subsets of A.

10. Sets A and B are disjoint if, and only if, _____.

they have no elements in common, or A \cap B = \emptyset.

11. A collection of nonempty sets A_1, A_2, A_3, \dots is a partition of a set A if, and only if, _____.

all A_i are a subset of A, but are also disjoint.

A is the union of all the sets A_1, A_2, A_3, \dots and A_i \cap A_j = \emptyset whenever i \neq j.

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Test Yourself

1. To prove that a set X is a subset of a set A \cap B, you suppose that x is any element of X and you show that x \in A _____ x \in B.

and

2. To prove that a set X is a subset of a set A \cup B, you suppose that x is any element of X and you show that x \in A _____ x \in B.

or

3. To prove that a set A \cup B is a subset of a set X, you start with any element x in A \cup B and consider the two cases _____ and _____. You then show that in either case _____.

x \in A; x \in B; x \in X

4. To prove that a set A \cap B is a subset of X, you suppose that _____ and you show that _____.

x \in A \cap B; x \in X

5. To prove that a set X equals a set Y, you prove that _____ and that _____.

X \subseteq Y; Y \subseteq X

6. To prove that a set X does not equal a set Y, you need to find an element that is in _____ and not _____ or that is in _____ and not _____.

X; in Y; Y; in X