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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Page 426

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

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Page 435

Test Yourself

1. Given a proposed set identity involving set variables A, B, and C, the most common way to show that the equation does not hold in general is to find concrete sets A, B, and C that, when substituted for the set variables in the equation, _____.

the equation does not hold.

make the left-hand side unequal to the right-hand side

2. When using the algebraic method for proving a set identity, it is important to _____ for every step.

cite the property from 6.2.2 used

3. When applying a property from Theorem 6.2.2, it must be used _____ as it is stated.

exactly

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Page 445

Test Yourself

1. In the comparison between the structure of the set of statement forms and the set of subsets of a universal set, the or operation \vee corresponds to _____, the and operation \wedge corresponds to _____, a tautology \mathbf{t} corresponds to _____, a contradiction \mathbf{c} corresponds to _____, and the negation operation, denoted \neg, corresponds to _____.

\cup;$\cap$,$U$,$\emptyset$,$^c$

2. The operations of + and \cdot in a Boolean algebra are generalizations of the operations of _____ and _____ in the set of all statement forms in a given finite number of variables and the operations of _____ and _____ in the set of all subsets of a given set.

\vee;$\wedge$;$\cup$;$\cap$

3. Russell showed that the following proposed "set definition" could not actually define a set: _____.

the set of all sets that are not elements of themselves