First formalize the following statement with logical symbols (this big question is *** 2 small questions, each with 2 points and * * * 4 points).

Everyone's fingerprints are different.

Analysis: Let M (x): X is a person; N (x, y): x ≠ y, that is, x and y are different people; E(x, y): the fingerprint phase of x and y.

Same. The original sentence can be formalized in one of the following two forms:

Note: If only N(x, y) is missing, only 1 is given.

2. Natural numbers are either odd or even, and odd numbers are not divisible by 2.

Analysis: Let P (x): x be a natural number, Q (x): x be an odd number, R (x): x be an even number, and D (x): x be divisible by 2.

The original sentence can be formalized as:

Note: (1) If only part of the content is answered correctly, the highest score is 1.

(2) The whole sentence must be written as a formula, with the conjunction ∧ connected in the middle, otherwise 0.5 points will be deducted.

2. Fill in the blanks (this big question is *** 4 small questions, 1 small questions are 1 points, and the second, third and fourth small questions are 2 points, *** 10 points)

1. Let A and B be finite sets, and the cardinality of A and B is m and n (m >; 0，n & gt0)。

(1) When m and n satisfy _ m = n _ _ _ _ There is a bijection function from a to b.

? At this time * * * can generate __ m! _____ ? Different bijection functions.

(2) When m and n satisfy _ _ _ _ _, there is an injective function from a to b.

At this time * * * can produce _ _ _ _ _? Different injective functions.

Analysis: This question can refer to the function part of mathematical formula set, a computer major with the same academic ability as Shen Shuo.

As we all know, five teachers and three students are sitting around the round table. If students are not adjacent, there are _ _ _1440 _ _ seating schemes.

Analysis: There are (5- 1) ways to make teachers sit together! = 4! =24, at this time, three students have five slots, which can ensure that students are not adjacent, so there are, so the total is 24*60= 1440.

3. Are there _ _16 _ _ odd numbers divisible by 23 10? A.

Analysis: The factorization set of 23 10 is {1, 2, 3, 5, 7, 1}, and the product of numbers other than1is the odd set A={3, 5, 7,/kloc. There are 15 species, and the combination of element 1 has 1 species, so there are 15+ 1= 16 species * * in total.

4. Let the vertex set of a graph be V(G)={} and the edge set be E(G)={}. Then the spanning tree of G has _ _ 8 _ _.

Analysis: draw the problem first, and then turn it into a spanning tree by breaking the circle. Delete can be divided into two parts, one part contains edges and the other part does not. Some of them include this edge and some don't, so there are 8 trees in total.

Iii. Answer the questions (this big question is *** 3 small questions, 1, 2 small questions are 4 points each, and the third small question is 8 points, *** 16 points).

1. Suppose ┐P, P∧Q, and P∨Q are expressed by the conjunction ↓ respectively.

Analysis:

( 1)- 1.

(2)

(3)

Only one question in (2) and (3) is correct, give 2 points, and if (2) and (3) are correct, give 3 points.

2. Let T be a tree with 13 vertices, and call this vertex with a moderate degree of 1 a leaf. If the degree of the vertex of T can only be 1, 2, 5 and T happens to have three vertices with degree 2, how many leaves does T have?

Analysis: If there are X leaves in T, then there are 13-3-x = 10-x in T? Vertex with degree 5,

Since the number of edges of a tree is equal to the number of vertices minus 1, the number of edges is 12? -Two points.

When the sum of vertices is equal to twice the number of edges, x+3 * 2+5 * (10-x) =12 * 2 = 24.

The solution is x = 8, so there are 8 leaves in it. -Two points.

3. To find the N-digit number consisting of five numbers: 1, 4, 5, 8 and 9, it is required that the number of occurrences of 4 and 8 is even, while the number of occurrences of 1, 5 and 9 is not limited.

Analysis: Let the number of I digits that meet the conditions be, then the exponential generating function corresponding to the sequence is? -Two points.

Because,

So-three points.

then what ? -Two points.

So what? ? - 1.

Iv. Proof questions (this big question is *** 2 small questions, the first 1 small question is 4 points, the second small question is 6 points, and *** 10 points).

1. Let r be a binary relation on nonempty set A, and r satisfies the following conditions:

(1)R is reflexive;

(2) If

Try to prove that r is an equivalent relation on a.

It is proved that R satisfies reflexivity from the condition (1). It is necessary to prove that r satisfies symmetry and transitivity.

1) for any

? & lta, b & gt∈R and conditions (1)

& lta，b & gt∈R∧& lt； a，a & gt∈R？ -1 3 points.

By condition (2)

Therefore, r satisfies symmetry.

2) For any

? & lta，b & gt∈R∧& lt； b，c & gt∈R

By symmetry? & ltb，a & gt∈R∧& lt； b，c & gt∈ R- 1。

By condition (2)

So r satisfies transitivity. Comprehensive 1), 2) available, r is the equivalent relation on a.

2. Paint each square of a 9X3 chessboard in red or blue at will, which proves that there must be two rows of squares with the same color.

It is proved that there is a way to paint 1x3 chessboard in red and blue. ? -Two points.

Let represent the first coloring method. Let any 9x3 chessboard that has been colored with red and blue be represented by the coloring method of the kth line. Settings? Principle of merger

And -2 points.

The nine elements in B are put in these eight drawers. According to the pigeon coop principle, there must be a positive integer, that is, at least two elements in B might as well be set together, which shows that the first and second lines of the chessboard are the same. -Two points.