Question 1

Decide whether the given set of numbers is a commutative group under the given operation.
-Rational numbers; multiplication

Accepted Answer

Rational numbers under multiplication do not form a group because the multiplicative inverse of 0 (a rational number) does not exist. For a set to be a group under a certain operation, every element must have an inverse within the set.

Question 2

Decide whether the given set of numbers is a group under the given operation.
-Even integers; multiplication

Accepted Answer

For a set to form a group under a given operation, it must satisfy four properties: closure, associativity, identity, and invertibility. The set of even integers under multiplication fails the identity and invertibility properties. There is no even integer that can act as a multiplicative identity (1 is the multiplicative identity but it is not an even integer), and not every even integer has a multiplicative inverse within the even integers (for example, 2 has no even integer as its multiplicative inverse since 1/2 is not an even integer).

Question 3

Decide whether the given set of numbers is a group under the given operation.
-Rational numbers; multiplication

Accepted Answer

Rational numbers under multiplication do not form a group because the multiplicative inverse of 0 (a rational number) does not exist. For a set to be a group under a certain operation, every element must have an inverse within the set.

Question 4

Decide whether the given set of numbers is a commutative group under the given operation.
-Even integers; multiplication

Accepted Answer

The set of even integers under multiplication is not a commutative group because it is not closed under the operation. For example, the product of two even integers is even, but the product of an even integer and an odd integer is odd, which is not in the set of even integers. Additionally, the set does not have an identity element since there is no even integer that can be multiplied by any even integer to give the same even integer.

Question 5

Decide whether the given set of numbers is a commutative group under the given operation.
-Natural numbers; addition

Accepted Answer

The set of natural numbers with addition is not a commutative group because it does not have an identity element (there is no natural number that can be added to any other natural number to get 0).

Question 6

Decide whether the given set of numbers is a commutative group under the given operation.
-Prime numbers; multiplication

Accepted Answer

Not all prime numbers have an inverse under multiplication. For example, there is no prime number that, when multiplied by 2, gives a whole number prime, so the set does not satisfy the inverse property of a group. Therefore, it cannot be a commutative group.

Question 7

Decide whether the given set of numbers is a commutative group under the given operation.
-Integers; addition

Accepted Answer

Integers under addition are a commutative group because addition is commutative (a+b=b+a), there exists an identity element (0) and every element has an inverse (-a).

Question 8

Decide whether the given set of numbers is a group under the given operation.
-Whole numbers; multiplication

Accepted Answer

The set of whole numbers is not a group under multiplication because it does not satisfy the requirement for the existence of an inverse for every element. For example, there is no whole number that can be multiplied by 2 (a whole number) to produce 1 (the multiplicative identity), as 1/2 is not a whole number.

Question 9

Decide whether the given set of numbers is a commutative group under the given operation.
-Whole numbers; addition

Accepted Answer

The set of whole numbers under addition is not a commutative group because it lacks the existence of inverses for every element. For example, there is no whole number that can be added to 3 to get 0 (the identity element for addition).

Question 10

Decide whether the given set of numbers is a commutative group under the given operation.
-Whole numbers; multiplication

Accepted Answer

The set of whole numbers under multiplication is not a commutative group because it does not satisfy the inverse property. Every whole number except 0 has a unique inverse under multiplication, except for 0 which does not have an inverse. Therefore, the set of whole numbers under multiplication is not a group.

Question 11

Decide whether the given set of numbers is a commutative group under the given operation.
-Natural numbers; subtraction

Accepted Answer

The natural numbers under subtraction do not form a commutative group because subtraction is not associative and some natural numbers do not have additive inverses within the set. For example, 1 - 2 = -1, which is not a natural number.

Question 12

Decide whether the given set of numbers is a commutative group under the given operation.
-Even integers; addition

Accepted Answer

The set of even integers under addition forms a commutative group.

To prove this, we need to check four properties:
1. Closure: The sum of any two even integers is even, so the set is closed under addition.
2. Associativity: Addition is associative, so this property holds.
3. Identity element: The identity element for addition is 0, which is also an even integer.
4. Inverse element: For any even integer a, -a is also even and their sum gives the identity element 0.

Additionally, addition is commutative for all integers, so this property holds for even integers too. Hence, the set of even integers under addition is a commutative group.

Question 13

Provide an appropriate response.
-Does the associative property hold for the set of real numbers under the operation of subtraction? Support your claim with an example.

Accepted Answer

The answer of Provide an appropriate response.
-Does the associative property...

Question 14

Provide an appropriate response.
-Does the associative property hold for the set of real numbers under the operation of division? Support your claim with an example.

Accepted Answer

The answer of Provide an appropriate response.
-Does the associative property...

Question 15

Decide whether the given set of numbers is a group under the given operation.
-Prime numbers; multiplication

Accepted Answer

For a set to form a group under a certain operation, it must satisfy four properties: closure, associativity, identity, and invertibility. The set of prime numbers under multiplication does not have an identity element within the set (the identity for multiplication is 1, which is not a prime number), and not every element has an inverse within the set that is also a prime number.

Question 16

Decide whether the given set of numbers is a group under the given operation.
-Integers; addition

Accepted Answer

The set of integers under addition forms a group because it satisfies the four group axioms: closure, associativity, identity (the identity element is 0), and every element has an inverse (for any integer a, its inverse is -a).

Question 17

Decide whether the given set of numbers is a group under the given operation.
-Natural numbers; addition

Accepted Answer

For a set to be a group under a given operation, it must satisfy four properties: closure, associativity, identity, and invertibility. The set of natural numbers under addition is closed and associative, and it has an identity element (0). However, it lacks invertibility since, for any natural number, there does not exist another natural number that can be added to it to yield the identity element (0). Therefore, the set of natural numbers under addition is not a group.

Question 18

Decide whether the given set of numbers is a group under the given operation.
-Natural numbers; subtraction

Accepted Answer

Subtraction is not a closed operation under natural numbers, meaning subtracting two natural numbers does not always result in a natural number (e.g., 1 - 2 = -1, which is not a natural number). Additionally, subtraction does not satisfy the associative property, which is a requirement for a set to be considered a group under a given operation.

Question 19

Decide whether the given set of numbers is a group under the given operation.
-Whole numbers; addition

Accepted Answer

For a set to form a group under a certain operation, it must satisfy four properties: closure, associativity, identity, and invertibility. While whole numbers under addition are closed and associative, and have an identity element (0), they lack invertibility. Not every whole number has an additive inverse (a number that can be added to it to yield zero) within the set of whole numbers. For example, there is no whole number that can be added to 1 to get 0, since the additive inverse of 1 is -1, which is not a whole number.

Question 20

Decide whether the given set of numbers is a group under the given operation.
-Even integers; addition

Accepted Answer

The set of even integers forms a group under addition because it satisfies the four group axioms: closure (the sum of any two even integers is even), associativity (addition is associative), identity (0 is an even integer that acts as the additive identity), and inverses (for every even integer, its negative is also an even integer and serves as its additive inverse).

Quiz 9: Geometry

Decide whether the given set of numbers is a commutative group under the given operation. -Rational numbers; multiplication

Decide whether the given set of numbers is a group under the given operation. -Even integers; multiplication

Decide whether the given set of numbers is a group under the given operation. -Rational numbers; multiplication

Decide whether the given set of numbers is a commutative group under the given operation. -Even integers; multiplication

Decide whether the given set of numbers is a commutative group under the given operation. -Natural numbers; addition

Decide whether the given set of numbers is a commutative group under the given operation. -Prime numbers; multiplication

Decide whether the given set of numbers is a commutative group under the given operation. -Integers; addition

Decide whether the given set of numbers is a group under the given operation. -Whole numbers; multiplication

Decide whether the given set of numbers is a commutative group under the given operation. -Whole numbers; addition

Decide whether the given set of numbers is a commutative group under the given operation. -Whole numbers; multiplication

Decide whether the given set of numbers is a commutative group under the given operation. -Natural numbers; subtraction

Decide whether the given set of numbers is a commutative group under the given operation. -Even integers; addition

Provide an appropriate response. -Does the associative property hold for the set of real numbers under the operation of subtraction? Support your claim with an example.

Provide an appropriate response. -Does the associative property hold for the set of real numbers under the operation of division? Support your claim with an example.

Decide whether the given set of numbers is a group under the given operation. -Prime numbers; multiplication

Decide whether the given set of numbers is a group under the given operation. -Integers; addition

Decide whether the given set of numbers is a group under the given operation. -Natural numbers; addition

Decide whether the given set of numbers is a group under the given operation. -Natural numbers; subtraction

Decide whether the given set of numbers is a group under the given operation. -Whole numbers; addition

Decide whether the given set of numbers is a group under the given operation. -Even integers; addition