The first problem is to verify the binding law, and the distribution rate is more troublesome. It will be more convenient to build a "model"

(but it may not be the standard method). Consider Z2.

=

{0, 1} is mod.

2, it is easy to know that Z2 is a field. Z2[x] is a univariate polynomial ring of Z2 coefficient, which can verify X? +x+ 1 is irreducible in Z2[x]. So x? Ideal generated by +x+ 1 in Z2[x] (x? +x+ 1) is the maximal ideal quotient ring Z2[x]/(x? +x+ 1) is a field. Quotient ring Z2[x]/(x? +x+ 1) has four elements, which are equivalent classes of 0, 1, x, x+ 1. You can write an operation table:+

1

x

x+ 10

1

x

x+ 1 1

1

x+ 1

xx

x

x+ 1

1x+ 1

x+ 1

x

1

0×

1

x

x+ 10

0 1

1

x

x+ 1x

x

x+ 1

1x+ 1

x+ 1

1

X establishes g and Z2[x]/(x? +x+ 1), 0, 1, 2,3 is mapped to the equivalence class to which 0, 1, x, x+ 1 belongs. It can be seen that this one-to-one correspondence keeps the operation, so it is determined by Z2[x]/(x? +x+ 1) is a domain. Of course, because it is more troublesome to ask (1), you can leave it alone (x? +x+ 1) is not ideal. It only needs Z2[x]/(x? +x+ 1) is just a ring, so it can be a ring. On this basis, it is easy to see the multiplication exchange in G from the multiplication table. The unit element is 1, and all non-zero elements are reversible, that is, fields. Note: 4 yuan finite fields are unique in the sense of isomorphism in principle, so since G is a 4 yuan finite field, it must be related to Z2[x]/(x? +x+ 1) isomorphism. ( 1)

A

=

{x

|

x

∈

z，x

& lt

0} and natural number set n

=

{x

|

x

∈

z，x

≥

0} can establish a one-to-one correspondence. F: A.

→

n，f(x)

=

-1-X. so the cardinal number of a is aleph0. (2)

B

=

(0, 1/2) and real number set r can establish a one-to-one correspondence. g:B

→

r，g(x)

=

Tan(π(2x- 1/2))。 So the cardinal number of b is aleph 1. I don't study discrete mathematics, so there may be inconsistencies in terms or theoretical expressions.