Website owner: James Miller
Examples of various algebraic systems
1. Let M be the set of all positive even integers i.e. the set 2, 4, 6, ... under the operation of ordinary addition. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has no identity element
- elements don’t have inverses
Thus it meets the axiomatic requirements for being a semigroup.
2. Let M be the set of all positive even integers plus the element 0 i.e. the set 0, 2, 4, 6, ... under the operation of ordinary addition. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has an identity element (the element 0)
- elements don’t have inverses
Thus it meets the axiomatic requirements for being a monoid.
3. Let M be the set of all even integers i.e. the set .... -6, -4, -2, 0, 2, 4, 6, .... under the operation of ordinary addition. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has an identity element (the element 0)
- every element has an inverse
Thus it meets the axiomatic requirements for being a group.
4. Let M be the set of all odd integers i.e. the set ... -5, -3, -1, 1, 3, 5, .... under the operation of ordinary addition. What algebraic properties does it have?
Answer.
- it is not closed
Because it is not closed, it does not constitute an algebraic system.
5. Let M be the set of all positive integers i.e. the set 1, 2, 3, 4, 5, 6, ... under the operation of ordinary addition. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has no identity element
- elements don’t have inverses
Thus it meets the axiomatic requirements for being a semigroup.
6. Let M be the set of all integers i.e. the set .... -3, -2, -1, 0, 1, 2, 3, .... under the operation of ordinary addition. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has an identity element (the element 0)
- each element has an inverse
Thus it meets the axiomatic requirements for being a group.
7. Let M be the set of all even integers i.e. the set .... -6, -4, -2, 0, 2, 4, 6, .... under the operation of ordinary multiplication. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it does not have an identity element
- elements do not have inverses
Thus it meets the axiomatic requirements for being a semigroup.
8. Let M be the set of all integers i.e. the set .... -3, -2, -1, 0, 1, 2, 3, .... under the operation of ordinary multiplication. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has an identity element (the element 1)
- elements do not have inverses
Thus it meets the axiomatic requirements for being a monoid.
9. Let M be the set of all rational numbers under the operation of ordinary addition. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has an identity element (the element 0)
- elements have inverses
Thus it meets the axiomatic requirements for being a group
10. Let M be the set of all rational numbers under the operation of ordinary multiplication. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has an identity element (the element 1)
- elements have inverses
Thus it meets the axiomatic requirements for being a group
Moreover,
- The set M of all real numbers under ordinary addition is a group.
- The set M of all real numbers under ordinary multiplication is a group.
- The set M of all complex numbers under ordinary addition is a group.
- The set M of all complex numbers under ordinary multiplication is a group.
11. Let M be the set of all n-vectors under the operation of vector addition. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has an identity element (the zero vector)
- elements have inverses
Thus it meets the axiomatic requirements for being a group
12. Let M be the set of all nxn matrices whose elements are real numbers under the operation of matrix addition. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has an identity element (the zero matrix)
- elements have inverses
Thus it meets the axiomatic requirements for being a group
13. Let M be the set of all nxn matrices whose elements are real numbers under the operation of matrix multiplication. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has an identity element (the identity matrix)
- elements may not have inverses
Thus it meets the axiomatic requirements for being a monoid
14. Let M be the set of all nonsingular nxn matrices whose elements are real numbers under the operation of matrix multiplication. What algebraic properties does it have?
Answer.
- it is closed
- the associative law holds
- it has an identity element (the identity matrix)
- elements have inverses
Thus it meets the axiomatic requirements for being a group
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