Indicate which of the following relationships are true and which are false:
a. [tex]$Z ^{+} \subseteq Q$[/tex]
b. [tex]$R \subseteq$[/tex]
c. [tex]$Q \subseteq Z$[/tex]
d. [tex]$Z , \cup Z = Z$[/tex]
e. [tex]$Z ^{-} \cap Z ^{+}=\emptyset$[/tex]
f. [tex]$Q \cap R = Q$[/tex]
g. [tex]$Q \cup Z=Q$[/tex]
h. [tex]$Z ^{ 4 } \cap R = Z ^{ 4 }$[/tex]
i. [tex]$Z \cup Q=Z$[/tex]
Answer (2)
The truth values of the statements about set relationships are as follows: a (True), b (False), c (False), d (True), e (True), f (True), g (True), h (True), and i (False). In summary, most statements affirm the relationships among the sets accurately, highlighting important properties of sets. Statement b is false due to its incompleteness, and statement i is false because it incorrectly conflates the union of integers and rational numbers with integers only.
;
Determine if the set of positive integers is a subset of the set of rational numbers: True.
Determine if the set of real numbers is a subset of an incomplete set: False.
Determine if the set of rational numbers is a subset of the set of integers: False.
Determine if the union of the set of integers with itself is the set of integers: True.
Determine if the intersection of the set of negative integers and the set of positive integers is the empty set: True.
Determine if the intersection of the set of rational numbers and the set of real numbers is the set of rational numbers: True.
Determine if the union of the set of rational numbers and the set of integers is the set of rational numbers: True.
Determine if the intersection of the set of integers raised to the power of 4 and the set of real numbers is the set of integers raised to the power of 4: True.
Determine if the union of the set of integers and the set of rational numbers is the set of integers: False.
Explanation
Problem Analysis We need to evaluate the truth value of the given set relationships. Let's analyze each statement.
Analyzing statement a a. Z + ⊆ Q : This statement means that the set of positive integers is a subset of the set of rational numbers. A rational number is any number that can be expressed as a fraction q p , where p and q are integers and q = 0 . Any positive integer n can be written as 1 n , which is a rational number. Therefore, this statement is true.
Analyzing statement b b. R ⊆ : This statement is incomplete and thus cannot be evaluated. Therefore, this statement is false.
Analyzing statement c c. Q ⊆ Z : This statement means that the set of rational numbers is a subset of the set of integers. However, not every rational number is an integer. For example, 2 1 is a rational number but not an integer. Therefore, this statement is false.
Analyzing statement d d. Z ∪ Z = Z : This statement means that the union of the set of integers with itself is the set of integers. The union of a set with itself is the set itself. Therefore, this statement is true.
Analyzing statement e e. Z − ∩ Z + = ∅ : This statement means that the intersection of the set of negative integers and the set of positive integers is the empty set. By definition, a number cannot be both negative and positive. Therefore, this statement is true.
Analyzing statement f f. Q ∩ R = Q : This statement means that the intersection of the set of rational numbers and the set of real numbers is the set of rational numbers. Since rational numbers are a subset of real numbers, their intersection is the set of rational numbers. Therefore, this statement is true.
Analyzing statement g g. Q ∪ Z = Q : This statement means that the union of the set of rational numbers and the set of integers is the set of rational numbers. Since integers are a subset of rational numbers, their union is the set of rational numbers. Therefore, this statement is true.
Analyzing statement h h. Z 4 ∩ R = Z 4 : This statement means that the intersection of the set of integers raised to the power of 4 and the set of real numbers is the set of integers raised to the power of 4. Since integers raised to the power of 4 are a subset of real numbers, their intersection is the set of integers raised to the power of 4. Therefore, this statement is true.
Analyzing statement i i. Z ∪ Q = Z : This statement means that the union of the set of integers and the set of rational numbers is the set of integers. Since integers are a subset of rational numbers, the union is the set of rational numbers, not the set of integers. Therefore, this statement is false.
Final Answer Here's the final answer: a. True b. False c. False d. True e. True f. True g. True h. True i. False
Examples
Understanding set relationships is fundamental in computer science, especially in database management and algorithm design. For example, when designing a database, knowing that the set of all customer IDs (integers) is a subset of all possible unique identifiers (rational numbers) helps in efficiently allocating storage and optimizing search queries. Similarly, in algorithm design, understanding set operations like union and intersection helps in creating efficient data structures and algorithms for tasks like data filtering and classification.
