A mathematical proof problem, a set.

Venn diagrams can be used to help analyze the meaning of the topic and clarify ideas; But take it as a proof process. There is a suspicion of lack of rigor. Below I give the algebraic proof process.

Proof: a b a

a∩b＜b（a∩b）^c＞a^c

a∩b）^c>b^c

a∩b）^c＞a^c∪b^c……※

The same can be argued, (a b) c a c b c

Substitute a c into a and b c into b so that there is.

a c b c) c (a c) c (b c) c=a b on both sides, get.

a^c∪b^c＞（a∩b）^c

i.e. (a b) c a c b c

The combination formula yields, :(a b) c = a c b c

First, b contains b c, so a b contains a (b c); Similarly, a c contains a (b c);

Thus (a b) (a c) contains a (b c);

Second, b c contains b, so a (b c) contains a b; Similarly, b c contains c, so a(b c) contains a c, so a(b c) contains (a b) (a c);

In summary, a (b c) = (a b) (a c).

Idea analysis: The meaning of the question is intuitively represented by a venn diagram, the conditions are intuitively expressed, and the problem is easily solved.

15-3-3 = 9 (people) who only participated in swimming competitions;

Those who only participate in track and field competitions have 8-3-x=5-x (people);

Those who only participate in ball games have 14-3-x = 11-x (people);

There are 3 people, 3 people, and X people who participated in the two competitions at the same time;

Therefore, 9+(5-x)+(11-x)+3+3+x=28, and the solution is x=3

Therefore, there are 3 people who participate in both track and field and ball games, and 9 people who only participate in swimming competition.

15 + (14-3) + (8-3) - 28 = 3 people So there are 3 people who participate in track and field and ball games at the same time.

15-3-3 = 9 people So there are only 9 people who participate in swimming.

This kind of inscription can be made with a Wayne diagram.

The easiest way to do this is to draw a picture.

If you make 4 circles, which represent the total participation, swimming, track and field, and ball, then the intersection of swimming and track and field is 3, the intersection of swimming and ball is 3, and the intersection of swimming, track and field, and ball is 0Then there is the following equation.

15 + 8 + 14-28 = 9 people, 9-3-3 = 3 people, this is the number of people participating in athletics and ball games.

28-8-14+3=9 people. This is the number of people who participate in swimming only.

Regarding the first question, since we are talking about a set, and the x in the set must satisfy the square of x plus 2 to equals 0, such an x does not exist, so such a set is an empty set. It's just an empty set, and it can't be said to be meaningless.

On the second question, the main difference between the two sets is the parentheses. In the first case, the numbers are covered with small parentheses, which means that the two numbers inside are the abscissa and ordinate of a certain point, so the first set element is the point (1,2), and the second set element is the point (2,1), which is naturally different. In the second case, there are no parentheses, indicating that the numbers in the set are all his elements, the first set has 4,5, and the second set has 5,4.

The two set elements are the same, so the set is naturally the same, regardless of the order of the elements.

x squared + 2 is the constant of 2, and it doesn't make sense to take any number.

The first sentence is true, x 2+2=0, this equation has no solution in the range of real numbers, because the square of any real number cannot be equal to -2, that is, it is meaningless in the range of real numbers.

The second sentence is wrong: notice that both sets are sets of points, and the points in the set are (1,2) and the other is (2,1) are not the same, and the elements are not the same for the set, so they do not represent a set.

The third sentence is true, both sets are numbers, 4, 5 and 5, 4 are no difference, so yes, I am a high school math teacher, can be in contact for a long time, if you have any questions, please direct me, I hope your grades improve.

In the range of real numbers is an empty set.

With are two different points and are different sets.

The same collection as represented.

The x-squared must be greater than 0. If x square 2 0, x square = i times the root number 2, this is an imaginary number.

It is the same set as the representation, for example, eating the main food first and eating the vegetables first are both called eating.

The same collection as the expression, such as putting water first and tea leaves first, is called making tea.

I have the impression that the definition field of the inverse proportional function is generally "and", and the use of "and" seems to be wrong.

For example, for example, y=1 x has a function image that decreases monotonically on (negative infinity, 0) and (0, positive infinity). This means that it meets the definition of monotonicity in two regions. Example:

take x1f(x2), so monotonically decreasing on (negative infinity, 0); Take 0f(x2), so monotonically decreasing on (0, positive infinity).

And if it is "and", it means that (negative infinity, 0) and (0, positive infinity) are regarded as an interval, and if you take x1<0 should be ......

The "and" must be a continuous plot.

and "can be two or even more paragraphs."

This question will be easy to do.

The derivative is evergreen at zero, so it increases.

FO F2 so use and choose C

n(m) indicates the number of elements in the set m.

n(m)=3, that is, the number of elements is 3.

x-5)(p-x)>0;i.e. (x-5)(x-p)<0;

If p<5; The solution of the inequality is: p5; The solution of the inequality is: 5

This means that the set m is greater than zero and px is an integer.

（x-5）(x-p)<0

5 The first case.

p=9, the second case.

p = 1, so p can only take 1 or 9

Choose A, the process hi me, I'll tell you.

x and y are both odd sets, x+y is an even number, m is an even set, so choose a

b is an option, m is an even set, x is an odd set, and y is a subset of an odd set, which is when k is an even number.

x is all odd y odd then x+y is an even number m represents all even numbers, so choose a

The number brought in can also be proved that if k=1 then, x=3, y=5 then x+y=8, then we can see that 8 is an even number, so m

Choosing b because x is seen as a set of odd numbers, and y is seen as a set of fours divided by 1 (which is certainly also part of an odd number).

Then y is a subset of x, additive, or x

a=3n+2

a 3=(3n+2) 3=n+2 Crack like regret 3, so a means divide by 3, the remainder is the set of positive certificates of 2.

Let k=m+1

then b = 3 (m + 1) - 1 = 3 m + 2

b/3=m+2/3

So b is also a set of certificates divided by 3 and the remainder is 2.

The rubber is referred to as a=b

Learn induction. 1) When n=1.

Divisible by 6.

When n=2.

Divisible by 6.

2) Suppose when n=k (k is a positive integer).

k 3-k is divisible by 6.

then when n=k+1.

k+1)^3-(k+1)=(k+1)[(k+1)^2-1]=(k+1)(k+2)k

k(k+1)(k+2) is the product of three consecutive positive integers.

There must be a multiple of 3 in three consecutive positive integers.

At least one is an even number.

Therefore, k(k+1)(k+2) has two messy factors, 2 and 3.

It must be able to be an integer of 6.

It can be seen from the comprehensive praise (1) and (2).

For any positive integer, the ratio n 3-2 is a multiple of 6.

A few more elements are both true and subset, and you can figure it out by figuring out the definition and doing more questions.