Notes on modern algebra and serialization of question types: Chapter 2 (properties of binary operation)

① Closure of binary operation （ Definition ）

All elements in the operation table are required to be in the set . requirement ： Judge whether a given binary operation satisfies the closeness .

② Commutativity of binary operations （ Definition ）

If a▲b=b▲a, The binary operation is said to be exchangeable , The elements in the operation table are symmetrical about the main diagonal . requirement ： Judge whether a given binary operation satisfies commutativity .

③ The combinability of binary operations （ Definition ）

If (a▲b)▲c=a▲(b▲c), It is said that the binary operation can be combined with . requirement ： Judge whether the given binary operation satisfies the associativity .

④ Allocability of binary operations （ Definition ）

If x▲(y+z)=x▲y+x▲z, said ▲ Yes + Left assignable ; If (y+z)▲x=y▲x+z▲x, said Yes + Right assignable , When the right and left can be allocated at the same time, it is called Yes + Distributable . requirement ： Judge whether the given binary operation satisfies the allocability .

⑤ Absorption law of binary operation （ Definition ）

x*(x+y)=x call * Satisfy the law of absorption . requirement ： Judge whether the given binary operation satisfies the absorption law .

⑥ Idempotent law and idempotent of binary operation （ Definition ）

If for any in the set x, There are x▲x=x, The operation is said to satisfy the idempotent law ; If not for any element , But only for some elements , Call this part of the element idempotent ; The elements on the main diagonal of the operation table satisfying the idempotent law are the same as the elements of the row and column header . requirement ： Determine whether an operation satisfies the idempotent law , Or find the idempotent of the operation in the set .
⑦ Elimination law of binary operation （ Definition ）： If x▲y=x▲z be y=z, So called Satisfy the left elimination law ; If yx=zx be y=z, said Satisfy right elimination law . requirement ： Determine whether an operation satisfies the elimination law .

Question type ：

1. set up A={x|x=2^n, among n∈N}, ask <A,×>、<A,+>,<A,/> Whether they are closed .
analysis ： This is a simple comprehension question , Examine the concept of operational closure . Closed by easy to know multiplication , Addition is not necessarily closed （ Such as 2+4=6, Not in the assembly ）, Division is not necessarily closed （ Such as 2/4）.

2. set up Q Is a set of rational numbers , yes Q Binary operations on , For any a,b∈Q,ab=a+b-ab, Ask if the operation is interchangeable ？
analysis ： This is a simple comprehension question , The concept of commutativity of operations . Start directly from the definition ,ba=b+a-ba=a+b-ab=ab, Therefore, it can be concluded that the operation is exchangeable .

3. set up A It's a non empty set ,▲ yes A Binary operations on , For any a,b∈A, There are a▲b=b, prove ▲ Is a associative operation .
analysis ： This is a simple comprehension question , Investigate the concept of operational associativity . By definition (a▲b)▲c=b▲c=c,a▲(b▲c)=a▲c=c, therefore (a▲b)▲c=a▲(b▲c), So the operation can be combined with .

4. In natural number set N On , Which of the following operations is not combined ？

analysis ：（ use ▲ Express * operation ）
① about A Options (a▲b)▲c=(a+b+3)▲c=(a+b+3)+c+3=a+b+c+6,a▲(b▲c)=a▲(b+c+3)=a+(b+c+3)+3=a+b+c+6, therefore (a▲b)▲c=a▲(b▲c), Meet the combination .
② about B Options , obviously 3 The smallest of the two natural numbers can be found in the first two numbers , Then compare the minimum value with the third number to find the smaller value , You can also find the minimum value in the last two numbers first , Then compare the minimum value with the first number to find the smaller value , The results are the same , Therefore, the operation satisfies the associativity .
③ about C Options ,(a▲b)▲c=(a+2b)▲c=(a+2b)+2c=a+2b+2c;a▲(b▲c)=a▲(b+2c)=a+2(b+2c)=a+2b+4c, Because the two formulas are not equal , Therefore, the operation does not satisfy the associativity .
④ about D Options , One conclusion to remember ： model N Multiplication is an operation that can be combined . According to this conclusion D Options meet associativity .

5. In natural number set N On , Which of the following operations can be combined ？

analysis ：（ use ▲ Express * operation ）
① about A Options ,(a▲b)▲c=(a-b)▲c=a-b-c, and a▲(b▲c)=a▲(b-c)=a-(b-c)=a-b+c, Because the two are not equal , Therefore, subtraction is not an associative operation ;
② about B Options , The proof method is similar to the minimum value of the previous question , It is easy to know that the maximum operation is a associative operation ;
③ about C Options ,(a▲b)▲c=||a-b|-c|, and a▲(b▲c)=|a-|b-c||, So the two are not necessarily equal , The operation is not associative ;
④ about D Options ,(a▲b)▲c=(|a|+|b|)▲c=|(|a|+|b|)|+|c|=|a|+|b|+|c|, and a▲(b▲c)=a▲(|b|+|c|)=|a|+|(|b|+|c|)|=|a|+|b|+|c|, Because the two are equal , Therefore, the operation can be combined with .

6.N Is a set of natural numbers , To any x,y∈N,x▲y=max{x,y},x+y=min{x,y}, prove ▲ and + Satisfy the law of absorption .
analysis ： This is a simple comprehension question , Investigation of the concept of the law of absorption . Start directly from the definition ,x▲(x+y)=x▲min{x,y}=max{x,min{x,y}}=x, therefore ▲ Satisfy the law of absorption ;x+(x▲y)=x+max{x,y}=min{x,max{x,y}}=x, therefore + It also satisfies the law of absorption , Certificate completion .

7. set up P(S) Is a collection S Power set of , stay P(S) Union of the intersection sum set of two binary operation sets defined on , Verify that the two operations satisfy the idempotent law .

analysis ： This is a simple comprehension question , Examine the concept of idempotent law . Start directly from the definition , Set arbitrary A∈P(S), be A∩A=A, Satisfy idempotent law ,A∪A=A, It also satisfies the idempotent law , Certificate completion .

8. set up <Z+,▲>, If to any x And arbitrary y∈Z+, Yes x▲y=LCM(x,y), Then the operation satisfies the following operation laws ？

analysis ： Understanding questions , Investigate the mastery of the properties of various binary operations . The least common multiple of two positive integers must be a positive integer , Therefore, it meets the requirements of closeness ; The exchange order does not affect the result of the least common multiple of two numbers , Therefore, it also meets the exchangeability ; The least common multiple of three numbers must be the least common multiple of the least common multiple of two groups of numbers , Therefore, no matter which group of numbers to find the least common multiple first , Therefore, it meets the combination ;x and y The least common multiple of and x and z It cannot be explained that the least common multiple of is equal y and z equal （ Such as 3 and 4 The least common multiple of and 3 and 12 The least common multiple of ）, Therefore, the elimination law is not satisfied ;x and x The least common multiple of is x, Therefore, it satisfies the idempotent law .

9. The four operations in the figure below , What satisfies exchangeability is

analysis ： Simple comprehension questions , Examine the understanding of Exchangeability . The elements in the operation table satisfying commutativity are symmetrical about the main diagonal , Thus, what can meet the exchangeability is * operation 、 + Operation and * operation .

10. The four operations in the right graph satisfy idempotency

analysis ： Simple comprehension questions , Examine the understanding of idempotency . The elements on the main diagonal of the operation table satisfying idempotency are the same as their row and column number elements , From this we can judge × Operation and * The operation satisfies idempotency .