Indicate which of the following relationships are true and which are false:
a. [tex]$Z ^{+} \subseteq Q$[/tex]
b. [tex]$R ^{-} \subseteq Q$[/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]$Z ^{+} \cap R = Z ^{+}$[/tex]
i. [tex]$Z \cup Q=Z$[/tex]
Answer (2)
In summary, the truth values of the statements are as follows: a. True, b. False, c. False, d. False, e. True, f. True, g. True, h. True, i. False. The relationships demonstrate the properties of positive integers, rational numbers, and real numbers clearly. The analysis helps in understanding how these sets interact with each other in mathematics.
;
Z + s u b se t e qQ : Every positive integer is a rational number.
R − s u b se t e qQ : Not every negative real number is a rational number (e.g., − 2 ).
Q s u b se t e qZ : Not every rational number is an integer (e.g., 2 1 ).
Z − c u p Z + = Z : The union of negative and positive integers does not include zero.
Z − c a p Z + = ∅ : Negative and positive integers have no common elements.
Q c a pR = Q : Rational numbers are a subset of real numbers.
Q c u pZ = Q : Integers are a subset of rational numbers.
Z + c a pR = Z + : Positive integers are a subset of real numbers.
Z c u pQ = Q : Integers are a subset of rational numbers.
Explanation
Problem Analysis We are asked to determine the truthfulness of several set relationships involving integers ( Z ), rational numbers ( Q ), and real numbers ( R ). We will analyze each statement individually, providing a justification for why it is true or false.
Analyzing statement a a. Z + ⊆ Q : This statement asks if every positive integer is a rational number. A rational number 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 fits the definition of a rational number. Therefore, this statement is true.
Analyzing statement b b. R − ⊆ Q : This statement asks if every negative real number is a rational number. Real numbers include both rational and irrational numbers. For example, − 2 is a negative real number, but it cannot be expressed as a fraction of two integers, so it is not a rational number. Therefore, this statement is false.
Analyzing statement c c. Q ⊆ Z : This statement asks if every rational number is an integer. Consider the rational number 2 1 . This is a rational number, but it is not an integer. Therefore, this statement is false.
Analyzing statement d d. Z − ∪ Z + = Z : This statement asks if the union of negative integers and positive integers equals all integers. The set of integers Z includes negative integers, positive integers, and zero. The union of Z − and Z + does not include zero. Therefore, this statement is false.
Analyzing statement e e. Z − ∩ Z + = ∅ : This statement asks if the intersection of negative integers and positive integers is the empty set. Negative integers are { ... , − 3 , − 2 , − 1 } and positive integers are { 1 , 2 , 3 , ... } . These two sets have no elements in common. Therefore, this statement is true.
Analyzing statement f f. Q ∩ R = Q : This statement asks if the intersection of rational numbers and real numbers equals the rational numbers. Since every rational number is also a real number, the intersection of the set of rational numbers and the set of real numbers is the set of rational numbers. Therefore, this statement is true.
Analyzing statement g g. Q ∪ Z = Q : This statement asks if the union of rational numbers and integers equals the rational numbers. Since every integer is also a rational number, the union of the set of rational numbers and the set of integers is the set of rational numbers. Therefore, this statement is true.
Analyzing statement h h. Z + ∩ R = Z + : This statement asks if the intersection of positive integers and real numbers equals the positive integers. Since every positive integer is also a real number, the intersection of the set of positive integers and the set of real numbers is the set of positive integers. Therefore, this statement is true.
Analyzing statement i i. Z ∪ Q = Z : This statement asks if the union of integers and rational numbers equals the integers. Since every integer is a rational number, the union of the set of integers and the set of rational numbers is the set of rational numbers, not the set of integers. Therefore, this statement is false.
Final Answer In summary: a. True b. False c. False d. False e. True f. True g. True h. True i. False
Examples
Understanding set relationships is crucial in computer science, especially in database management and algorithm design. For instance, when designing a database, knowing that the set of all possible integer IDs ( Z ) is a subset of all possible numeric IDs ( Q or R ) helps in defining data types and constraints. Similarly, in algorithm design, understanding set operations like union and intersection helps in optimizing search and sorting algorithms. These concepts ensure data integrity and efficient processing.
