13) Indicate which of the following relationships are true and which are false:
a. [tex]$Z ^{\prime} \subseteq Q$[/tex]
b. [tex]$R \subseteq$[/tex]
c. [tex]$Q \subseteq Z$[/tex]
d. [tex]$Z \cup Z=Z$[/tex]
e. [tex]$Z^{-} \cap Z^{+}=n$[/tex]
f. [tex]$Q \cap R = Q$[/tex]
g. [tex]$Q \cup Z=Q$[/tex]
h. [tex]$Z ^{\prime} \cap R = Z ^{+}$[/tex]
i. [tex]$Z \cup Q=Z$[/tex]
Answer (2)
The truth values for the given relationships are: a: False, b: False, c: False, d: True, e: True, f: True, g: True, h: False, i: False. Each set is analyzed based on their definitions in mathematics. Overall, the majority of set relationships assessed are correct, revealing the connections between rational, irrational, and integer numbers.
;
Z ′ ⊆ Q : False, irrational numbers are not rational.
R ⊆ : False, the statement is incomplete.
Q ⊆ Z : False, rational numbers include fractions.
Z ∪ Z = Z : True, union of integers with itself is integers.
Z − ∩ Z + = ∅ : True, negative and positive integers have no intersection.
Q ∩ R = Q : True, rational numbers are a subset of real numbers.
Q ∪ Z = Q : True, integers are a subset of rational numbers.
Z ′ ∩ R = Z + : False, intersection of irrational and real numbers is irrational numbers.
Z ∪ Q = Z : False, union of integers and rational numbers is rational numbers.
The final answers are: False, False, False, True, True, True, True, False, False.
Explanation
Analyzing the Statements We are given several statements about set relationships and need to determine if each is true or false. Here's a breakdown of each statement:
a. Z ′ ⊆ Q : This statement says that the set of irrational numbers is a subset of the set of rational numbers. This is false because irrational numbers (like 2 or π ) cannot be expressed as a fraction of two integers, which is the definition of rational numbers.
b. R ⊆ : This statement is incomplete. It seems to suggest that the set of real numbers is a subset of some undefined set. Since the set is not defined, we cannot determine if this is true or false. We will assume it is false due to incompleteness.
c. Q ⊆ Z : This statement says that the set of rational numbers is a subset of the set of integers. This is false because rational numbers can be fractions (like 2 1 ), which are not integers.
d. Z ∪ Z = Z : This statement says that the union of the set of integers with itself is equal to the set of integers. This is true because combining the set of integers with itself doesn't add any new elements.
e. Z − ∩ Z + = n : This statement says that the intersection of the set of negative integers and the set of positive integers is equal to n . We assume n represents the empty set ∅ . This is true because negative and positive integers have no common elements.
f. Q ∩ R = Q : This statement says that the intersection of the set of rational numbers and the set of real numbers is the set of rational numbers. This is true because rational numbers are a subset of real numbers, so their intersection is the set of rational numbers.
g. Q ∪ Z = Q : This statement says that the union of the set of rational numbers and the set of integers is the set of rational numbers. This is true because integers are a subset of rational numbers, so their union is the set of rational numbers.
h. Z ′ ∩ R = Z + : This statement says that the intersection of the set of irrational numbers and the set of real numbers is the set of positive integers. This is false because the intersection of irrational and real numbers is the set of irrational numbers, not positive integers.
i. Z ∪ Q = Z : This statement says that the union of the set of integers and the set of rational numbers is the set of integers. This is false because the union of integers and rational numbers is the set of rational numbers, since integers are a subset of rational numbers.
Determining Truth Values a. Z ′ ⊆ Q : False b. R ⊆ : False (incomplete statement) c. Q ⊆ Z : False d. Z ∪ Z = Z : True e. Z − ∩ Z + = ∅ : True f. Q ∩ R = Q : True g. Q ∪ Z = Q : True h. Z ′ ∩ R = Z + : False i. Z ∪ Q = Z : False
Final Truth Values The truth values of the given statements are: a. False b. False c. False d. True e. True f. True g. True h. False i. False
Examples
Understanding set theory is crucial in computer science, especially in database management and algorithm design. For instance, when designing a database, knowing the relationships between different sets of data (like integers, real numbers, and rational numbers) helps in optimizing data storage and retrieval. Similarly, in algorithm design, understanding set operations like union and intersection is essential for tasks such as data filtering and classification. These concepts ensure efficient and accurate data processing in various applications.
