
The set of all integral multiples of 5 is a subgroup of
a) The set of all rational numbers under multiplication
b) The set of all integers under multiplication
c) The set of all non-zero rational numbers under multiplication
d) The set of all integers under addition
Answer
485.1k+ views
Hint: A subgroup must satisfy all the properties of a group: Closure, Associativity, Identity, and Inverse. A sufficient condition is that it must satisfy the closure property. Note that “The set of all integral multiples of 5” is obtained when we multiply 5 with any number ‘n’.
Complete step-by-step solution:
Example: When we multiply 5 with , we get 5. When we multiply 5 with , we get 10 and so on.
A group is a set G together with an operation (.), which satisfies the following properties: If for and , we have that
Closure: For , we have . This says that for any two elements in G if we apply the operation (.) then the resulting element will also be in G
Associativity: For and , we have . This says that for any three elements in G, associativity between them holds when the operation (.) is applied.
Identity: For there exists an element such that , where e is the identity element.
Inverse: For there exists an element such that . Here is the inverse element.
A subgroup must satisfy all the properties of a group. The sufficient condition for a subgroup is that it must satisfy the closure condition.
“The set of all integral multiples of 5” is obtained when we multiply 5 with any number ‘n’. Let . Here ‘I’ is a set with all ‘n’ values. Here H is a non-empty set with at least one element.
Let and . Now we apply an addition operator for a and b. So, . Since and , . Hence, closure is satisfied.
Hence, the correct answer is
The set of all integral multiples of 5 is a subgroup of
d)The set of all integers under addition.
Note: We can also find the correct answer by applying all the properties of a group to the set of all integral multiples of 5. Example: let , and . Now , , , where , and , where
Hence, the set of all integral multiples of 5 satisfies all the properties of a group. This verifies our answer that the set of all integral multiples of 5 is a subgroup of the set of all integers under addition.
Complete step-by-step solution:
Example: When we multiply 5 with
A group is a set G together with an operation (.), which satisfies the following properties: If for
Closure: For
Associativity: For
Identity: For
Inverse: For
A subgroup must satisfy all the properties of a group. The sufficient condition for a subgroup is that it must satisfy the closure condition.
“The set of all integral multiples of 5” is obtained when we multiply 5 with any number ‘n’. Let
Let
Hence, the correct answer is
The set of all integral multiples of 5 is a subgroup of
d)The set of all integers under addition.
Note: We can also find the correct answer by applying all the properties of a group to the set of all integral multiples of 5. Example: let
Hence, the set of all integral multiples of 5 satisfies all the properties of a group. This verifies our answer that the set of all integral multiples of 5 is a subgroup of the set of all integers under addition.
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