additive inverse associative commutative multiplicative cancellation additive identity distributive closure There are also three essential properties of the equality numbers reflexive symmetric transitive Finally, substitution plays a role in both sets of properties. In each of the boxes below enter one of the phrases from the list given above which indicates the correct nam of the words in the list or put more than one space between the words in a phrase. For all x, y, z E Z х, у € Z, х+у€ Z · (y• z) = (x · y) · z There is a number i such that for all x, x + i = x x + z = z + x a = b ^ b = c а 3 с For x + 0, rx = sx → r =S a = b → a+c= b+ c (y + z) · x = y · x + z · x x = x a = b → b = a With i naming the identity element, for every y E Z there is a yop such that a + aop = i
additive inverse associative commutative multiplicative cancellation additive identity distributive closure There are also three essential properties of the equality numbers reflexive symmetric transitive Finally, substitution plays a role in both sets of properties. In each of the boxes below enter one of the phrases from the list given above which indicates the correct nam of the words in the list or put more than one space between the words in a phrase. For all x, y, z E Z х, у € Z, х+у€ Z · (y• z) = (x · y) · z There is a number i such that for all x, x + i = x x + z = z + x a = b ^ b = c а 3 с For x + 0, rx = sx → r =S a = b → a+c= b+ c (y + z) · x = y · x + z · x x = x a = b → b = a With i naming the identity element, for every y E Z there is a yop such that a + aop = i
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Mathematical Properties and Their Applications**
The following concepts are fundamental to understanding various properties in mathematics:
- **Properties:**
- Additive Inverse
- Associative
- Commutative
- Multiplicative Cancellation
- Additive Identity
- Distributive
- Closure
- **Equality Properties:**
- Reflexive
- Symmetric
- Transitive
Additionally, **substitution** is important in both groups of properties.
**Instructions:**
For each mathematical statement, choose the correct property from the lists above. Write the property name in the corresponding box without spacing between words in a phrase.
1. **Statement:**
\[
\text{For all } x, y, z \in \mathbb{Z}
\]
2. **Statement:**
\[
x, y \in \mathbb{Z}, \quad x + y \in \mathbb{Z}
\]
3. **Statement:**
\[
x \cdot (y \cdot z) = (x \cdot y) \cdot z
\]
4. **Statement:**
\[
\text{There is a number } i \text{ such that for all } x, \quad x + i = x
\]
5. **Statement:**
\[
x + z = z + x
\]
6. **Statement:**
\[
a = b \land b = c \quad \longrightarrow \quad a = c
\]
7. **Statement:**
\[
\text{For } x \neq 0, \quad rx = sx \quad \longrightarrow \quad r = s
\]
8. **Statement:**
\[
a = b \quad \longrightarrow \quad a + c = b + c
\]
9. **Statement:**
\[
(y + z) \cdot x = y \cdot x + z \cdot x
\]
10. **Statement:**
\[
x = x
\]
11. **Statement:**
\[
a = b \quad \longrightarrow \quad b = a
\]
12. **Statement:**
\[
\text{With }](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd84c1829-b5be-42b1-ba72-3bae7a546286%2F30ea2077-6c31-4b3b-9cab-4aa5c206bd74%2F10k8knz_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematical Properties and Their Applications**
The following concepts are fundamental to understanding various properties in mathematics:
- **Properties:**
- Additive Inverse
- Associative
- Commutative
- Multiplicative Cancellation
- Additive Identity
- Distributive
- Closure
- **Equality Properties:**
- Reflexive
- Symmetric
- Transitive
Additionally, **substitution** is important in both groups of properties.
**Instructions:**
For each mathematical statement, choose the correct property from the lists above. Write the property name in the corresponding box without spacing between words in a phrase.
1. **Statement:**
\[
\text{For all } x, y, z \in \mathbb{Z}
\]
2. **Statement:**
\[
x, y \in \mathbb{Z}, \quad x + y \in \mathbb{Z}
\]
3. **Statement:**
\[
x \cdot (y \cdot z) = (x \cdot y) \cdot z
\]
4. **Statement:**
\[
\text{There is a number } i \text{ such that for all } x, \quad x + i = x
\]
5. **Statement:**
\[
x + z = z + x
\]
6. **Statement:**
\[
a = b \land b = c \quad \longrightarrow \quad a = c
\]
7. **Statement:**
\[
\text{For } x \neq 0, \quad rx = sx \quad \longrightarrow \quad r = s
\]
8. **Statement:**
\[
a = b \quad \longrightarrow \quad a + c = b + c
\]
9. **Statement:**
\[
(y + z) \cdot x = y \cdot x + z \cdot x
\]
10. **Statement:**
\[
x = x
\]
11. **Statement:**
\[
a = b \quad \longrightarrow \quad b = a
\]
12. **Statement:**
\[
\text{With }
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