8 Questions
What is the multiplicative identity in the real number system?
1
Which property of real numbers is demonstrated by the equation $(a imes b) imes c = a imes (b imes c)$?
Associativity
According to the trichotomy law, for any two real numbers $a$ and $b$:
Exactly one of the following is true: $a < b$, $a = b$, or $a > b$
Which of these is an example of the distributive property?
$a imes (b + c) = a imes b + a imes c$
What does the completeness axiom guarantee in a set of real numbers?
Every nonempty set of real numbers that is bounded above has a least upper bound (supremum)
Which of the following statements illustrates the total order property?
For any two real numbers $a$ and $b$, exactly one of these is true: $a < b$, $a = b$, or $a > b$
What does transitivity mean in the context of real numbers?
If $a eq b$, then $a = c$
Which statement best describes the additive inverse property?
For any $a$, there exists $-a$ such that $a + (-a) = 0$
Study Notes
Real Numbers Systems
Definition
- A real number system is a mathematical structure consisting of a set of real numbers, together with operations of addition and multiplication, and an ordering relation.
Properties
-
Commutativity:
- Addition: a + b = b + a
- Multiplication: a × b = b × a
-
Associativity:
- Addition: (a + b) + c = a + (b + c)
- Multiplication: (a × b) × c = a × (b × c)
- Distributivity: a × (b + c) = a × b + a × c
-
Existence of additive and multiplicative identities:
- Additive identity: 0 (a + 0 = a)
- Multiplicative identity: 1 (a × 1 = a)
-
Existence of additive inverses:
- For each a, there exists -a such that a + (-a) = 0
-
Existence of multiplicative inverses:
- For each a ≠ 0, there exists 1/a such that a × (1/a) = 1
Ordering Relation
- Total Order: For any two real numbers a and b, exactly one of the following is true: a < b, a = b, or a > b
- Trichotomy Law: For any two real numbers a and b, a ≤ b or b ≤ a
- Transitivity: If a ≤ b and b ≤ c, then a ≤ c
Completeness Axiom
- Least Upper Bound Property: Every nonempty set of real numbers that is bounded above has a least upper bound (supremum)
- Greatest Lower Bound Property: Every nonempty set of real numbers that is bounded below has a greatest lower bound (infimum)
Real Numbers Systems
Definition
- A real number system is a mathematical structure consisting of a set of real numbers, together with operations of addition and multiplication, and an ordering relation.
Properties
- Commutative Property: • Addition: the order of numbers does not change the result (a + b = b + a) • Multiplication: the order of numbers does not change the result (a × b = b × a)
- Associative Property: • Addition: the order in which numbers are added does not change the result ((a + b) + c = a + (b + c)) • Multiplication: the order in which numbers are multiplied does not change the result ((a × b) × c = a × (b × c))
- Distributive Property: multiplication is distributed over addition (a × (b + c) = a × b + a × c)
- Existence of Identities: • Additive identity: 0 (a + 0 = a) • Multiplicative identity: 1 (a × 1 = a)
- Existence of Inverses: • Additive inverse: for each a, there exists -a such that a + (-a) = 0 • Multiplicative inverse: for each a ≠ 0, there exists 1/a such that a × (1/a) = 1
Ordering Relation
- Total Order: exactly one of the following is true: a < b, a = b, or a > b
- Trichotomy Law: for any two real numbers a and b, a ≤ b or b ≤ a
- Transitivity: if a ≤ b and b ≤ c, then a ≤ c
Completeness Axiom
- Least Upper Bound Property: every nonempty set of real numbers that is bounded above has a least upper bound (supremum)
- Greatest Lower Bound Property: every nonempty set of real numbers that is bounded below has a greatest lower bound (infimum)
This quiz covers the definition and properties of real number systems, including commutativity, associativity, and distributivity.
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