explain why are the following not groups:
1)the set of Z integers with operation subtraction
2)The set of Z integers with operation addition
3)The set of R* of all non zero real numbers with addition
1) The set of integers, Z, with the operation of subtraction is not a group because it does not satisfy the group properties. In order for a set with an operation to be a group, it must have closure, associativity, identity, and inverses.
Closure means that when you perform the operation on any two elements in the set, the result is also in the set. In the set of integers with subtraction, closure is not satisfied because subtracting two integers may result in a non-integer, such as a decimal or a fraction.
Associativity means that the order in which you perform the operation does not matter. In the case of subtraction with integers, this property is satisfied.
Identity means that there exists an element in the set that, when combined with any other element, does not change that element. In the set of integers with subtraction, there is no identity element because there is no integer, when subtracted from another integer, that will give back the original integer.
Inverses mean that for every element in the set, there exists an element that, when combined with the original element, gives the identity element. In the set of integers with subtraction, not every element has an inverse because the result of subtracting any integer from 0 may not be an integer.
Therefore, the set of integers with subtraction does not satisfy the group properties and is not a group.
2) The set of integers, Z, with the operation of addition is a group. Here's why:
- Closure: When you add any two integers, the result is also an integer, so closure is satisfied in this case.
- Associativity: The order in which you add integers does not change the result, so associativity is satisfied.
- Identity: The identity element is 0, because adding 0 to any integer gives back the original integer.
- Inverses: For every integer, there exists an inverse that, when added to the original integer, gives the identity element (0). For example, the inverse of 5 is -5, and adding them together results in 0.
Therefore, the set of integers with addition satisfies all the group properties and is a group.
3) The set of non-zero real numbers, R*, with the operation of addition is not a group. Here's why:
- Closure: When you add two non-zero real numbers, the result is still a non-zero real number, so closure is satisfied.
- Associativity: The order in which you add real numbers does not change the result, so associativity is satisfied.
- Identity: The identity element is 0, because adding 0 to any non-zero real number gives back the original number.
- Inverses: For every non-zero real number, there exists an inverse that, when added to the original number, gives the identity element (0). For example, the inverse of 5 is -5, and adding them together results in 0.
However, the set of non-zero real numbers with addition fails to satisfy the group requirement of having an identity element. The identity element must be an element that, when combined with any other element, does not change that element. In this case, the identity element is 0, which is not a member of the set of non-zero real numbers. Therefore, the set of non-zero real numbers with addition is not a group.
1) The set of integers Z with the operation of subtraction is not a group because it does not satisfy the group axioms.
- Closure: The set of integers is closed under subtraction since subtracting two integers will always yield another integer.
- Identity: In a group, there must be an identity element that, when combined with any element, gives that element back. However, in the set of integers with subtraction, there is no integer that can be subtracted from any other integer to get the same integer. Therefore, it lacks an identity element.
- Inverses: For every element in a group, there must exist an inverse element such that when they are combined, the identity element is obtained. However, in the set of integers with subtraction, there are integers that do not have an additive inverse within the set. For example, if we consider the integer 3, there is no integer that we can subtract from 3 to get the identity element, 0.
- Associativity: The operation of subtraction is associative on the set of integers. When subtracting three integers, the order does not matter.
Since the set of integers with subtraction does not have an identity element and inverses for all elements, it fails to satisfy the group axioms, and thus, it is not a group.
2) The set of integers Z with the operation of addition is a group.
- Closure: The set of integers is closed under addition since adding two integers will always yield another integer.
- Identity: The identity element in the set of integers with addition is 0. Adding any integer with 0 will give the original integer back.
- Inverses: For every element in the set, there exists an inverse element. For example, the inverse of the integer 3 is -3, and when added together, they give the identity element 0.
- Associativity: The operation of addition is associative on the set of integers.
Thus, the set of integers with addition satisfies all the group axioms and is a group.
3) The set of non-zero real numbers R* with addition is not a group.
- Closure: The set of non-zero real numbers is closed under addition since adding two non-zero real numbers will always yield another non-zero real number.
- Identity: The identity element in the set of non-zero real numbers with addition is missing. There is no non-zero real number that, when added with any other non-zero real number, will give back the original non-zero real number. The identity element would be 0, but it is excluded from the set.
- Inverses: For every element in the set, there exists an inverse element. For any non-zero real number, its additive inverse is the negation of the number, which is still in the set of non-zero real numbers.
- Associativity: The operation of addition is associative on the set of non-zero real numbers.
Although the set of non-zero real numbers with addition satisfies closure, inverses, and associativity, it lacks an identity element, which violates one of the group axioms. Therefore, it is not a group.