Notes on modern algebra and series of questions: Chapter 1 (introduction of algebraic system)

① A collection of n Meta operational （ Definition ）

For a non empty set , Take from it n Mapping elements , The result of the mapping is an element in the collection .
requirement ： Can judge whether an operation is a set of n Yuan operation （ First, judge whether the set is empty , Then judge its n Whether all elements and mapping results belong to the collection ）

② Operational closure （ Definition ）

If the operation result is still in the same set as the operands involved in the operation, it is closed .
requirement ： Can judge whether a certain operation satisfies the closeness .

③ Algebraic systems （ Definition ）

A whole consisting of a nonempty set and the operations defined on it .
requirement ： Able to understand , And can list the operation table according to the meaning of the question .

④ Operation table （ Relevant concepts ）

A table representing the result of a unary or binary operation .
requirement ： Be able to understand the operation table and analyze .

⑤ Algebraic systems of the same type （ Definition ）

The number of operations of the two algebraic systems is the same , The number of corresponding operations is also the same , And the number of algebraic constants is the same .
requirement ： Can judge whether two algebraic systems are of the same type .

Question type ：

1.Mn A unary operation defined on the can be ？

analysis ： Concept questions , Investigate the concept of operations on sets .Mn A unary operation on can be transpose 、 Take the opposite number of each element and so on , But it can't be an inverse operation , Because the inverse operation requires this n The determinant of order real matrix is not zero .

2. set up B={1,2}, Write P(B) Binary operation table and ~ Operation table of operation

analysis ： Simple comprehension questions , Investigate the production of operation table .P(B) It means B Power set of , by B Set of subsets of . The line number in the binary operation table represents the left operator , The column number represents the right operator , Fill in the operation results in the corresponding matrix ; The row number of the unary operation table indicates the operation element . The result of this question is as follows ：

3. The following three algebraic systems are of the same kind （）

analysis ： Concept questions , Investigate the concept of the same kind of algebraic system . The same kind of algebraic system requires the same number of operations in the algebraic system 、 The number of algebraic constants is the same and the number of corresponding operations is the same , Check one by one . The answer should be A.

4. Which of the following operations are not closed （）

analysis ： Simple comprehension questions , Examine the concept of operational closure . The subtraction of natural numbers may be negative , Does not satisfy the closeness ; Integer division may be decimal , Does not satisfy the closeness ;C The operation result of option may be 0,0 Don't set A in , Does not satisfy the closeness , So the answer is ABC.

5. set up A={1,2,3…10}, Then which of the following binary operations * It's closed （）

analysis ： Simple comprehension questions , Examine the concept of operational closure . The maximum or minimum value of two numbers naturally belongs to the set of these two numbers , Satisfy closure ; The greatest common divisor must belong to the set , Satisfy closure ; The least common multiple does not necessarily belong to the set , Such as 9 and 10 The least common multiple of 90. So the answer to this question is ABC.

6. For operation + - * |x-y| max min |x|, Which of the following sets are closed for this operation ？

analysis ： Simple comprehension questions , Examine the concept of operational closure .
① For addition operations , Integer addition 、 Natural number addition 、 Even addition belongs to the original set , But for CD Set does not hold , The options are ABE;
② For subtraction , Both integer subtraction and even subtraction belong to the original set , But for natural numbers and CD Sets do not necessarily hold , So the option is AE;
③ For multiplication , Integer multiplication 、 Natural number multiplication 、 Even multiplication belongs to the original set , But for CD Sets do not necessarily hold , So the option is ABE;
④ For the subtraction of absolute values , Set of integers 、 Set of natural numbers 、C Both option sets and even sets satisfy the closeness , however D Option set does not satisfy , So the option is ABCE;
⑤ For maximum and minimum operations , All sets satisfy closure , The options are ABCDE; ⑥ For absolute value operation , All sets satisfy closure , The options are ABCDE.