DIPLOMSKI - Uloga i Znacaj Skupova u Pocetnoj Nastavi Matematike

  • View
    858

  • Download
    14

Embed Size (px)

Text of DIPLOMSKI - Uloga i Znacaj Skupova u Pocetnoj Nastavi Matematike

................................................................................................................................4 1. ..............................................................8 1.1. .....................................................................................................8 1.2. ............................................................................................11 1.2.1. ...............................................................................14 1.2.2. .........................................................................17 1.2.3. () / /....18 1.2.4. ..........................................................................23 1.3. .............................................................................24 1.3.1. ................................................................................24 1.3.2. ............................................................................26 1.3.3. , ..........................................27 1.3.4. ( )...................................28 2. ..............................................................31 2.1. ................................................................................31 2.2. .................................................................................31 2.3. ....................................................................31 2.4. ....................................................................................31 2.5. ...............................................................................32 2.6. ..............................................................................32 2.7. .....................................................................33 2.8. ......................................................33 2.9. ...........................................................34 2.10. ............................................................34 2.11. ..................................................................35 3. ........................................................................36 3.1. ............................36

2

3.2. .....................................38 3.3. ........................................................................39 ................................................................................................................41 .............................................................................................................43 .....................................................................................................................44 1. .............................44 2. .............................................46

3

, , . UNESCO, 1956. . , , , , . , , , , . . , : episteme mathema. , , , . mathema mathematika ( ), . , , , , , . , , , , , . . . .

4

, . XIX , . , . , ( ) . , , , . . , . . , , . , , . ( ). ( ) , . , , , . , . , , , , : , . , 5

, , . . ( , ...) ( ) . . , . , . , . , , , . ; ; ( ). , , : , . . , . : ? . . , , , 6

. , . , , , . , . , . , .

7

1.

1.1. . . , , . , , , , , . , , . , : ( ), ( ), ( ) , , : , , , , . , . , . , , , ., - . . . . . : , , . . 8

. , . , (1848 1918), , , (1871 1956). ( ), . , : - , . - . - . . . . , , , . , . , 100 . , . , , , , , 1881. . , , , XVIII . , ; 9

. , , . :

:

*** , , , , ; je je . Ha , (3) (3) , je ( ). , , .

10

1.2. . . . , , . , , , . , B, ......X, Y,..., , b,...,x, y,... x X, xX, X, xX. : x X x X. xX x X x X, , . : , ? . . : k=0. , , , . : {x | S(x)} , , {x | x S}, x S. , X = {7, 8, 9, 10, 11, 12} : X={x | x N 6< x A C A = {1, 2, 3} B = {a, b, c}. B 1. 2. 3. A B, . 1 B, 2 b B 3 c B. : (1, a), (2, b), (3, c). , A B , A B . B B ( ). . a b c

19

. . k, , k(A) = k(B). B k(A) = k(B) B. 1. 2. 3. 4. . k (a) k (x) k(x) = 1. 0 k () = 0. . k (N) = N0 ( ). N0 . 5. () , .:

* * * *

* *

* * *

* *

1: A={a, e, , o, u }, |A| = 5 k(A) = 5 2: . . .

20

, . (3).

*** . , , . . , . , . , . . : , , . .

. , . , ,

21

, . , , ( ). .

. , , , , , ... , , . : , . , , . , : , , . , : , . , : , , . , : , . , ,

22

, . . , , , . . . , . . , . , . : , , , , . , : ? ( ). ? ? ? , , , . . , , (, ...). , , . , .

23

1.2.4. . . :

. . , N= {1, 2, 3, ...}, N0= {0, 1, 2, 3, ...}.

( , ). = {..., -2, -1, 0, 1, 2, ...}.

(. ). ={ : a,b b 0}. ,

. .

. (. , 2).

. , ,

. , . () ( ). , ,

( ),

P, N, Z, Q, C.

, , 3, 4, . R

24

1.3. . : , , , . 1.3.1. A B A B A B A B. A B = {x | x A x B} : 1) A B= B A 2) A A B B A B 3) A = A 4) A u = u

A

B

C

=

C=AB *** . , , . , . ,

25

B. B. C. C B, . C = B : UB B

, . . . , . . . , , . , , . , . 1: A={1, 2, 3, 4} B={3, 4, 5, 6, 7} AUB={1, 2, 3, 4, 5, 6, 7}

26

2: A = {, , } B = { , , , } AUB = {, , , , , , } BUA = {, , , ,, , }

A B

AUB

1.3.2. B. B. A B = {x | x A x } :

: A={1,2,3,4,5} B={4,5,6,7} A B = {4,5} B A = {4,5} AB A 1 2 3 4 5 5. 6 7 27 B

A B = B A. B , . B = . :

AB=BA A B A AA=A A=

: : A={1, 2, 3, 4}, B={3, 4, a, b} C={a, b, c, d} : 1) A B, 2) B C, 3) A C : 1) A B = {3, 4} 2) B C= {a, b} 3) A C =

1.3.3. , B , B: A/B = { x | x A x } B B, : B/A = { x | x B xA}, , . : A/B B/A : A= {3, 4, 5, 6} B= {4, 5, 7, 8} A/B = {3, 6} B/A = {7, 8}

28

A B 3 6 4 5 7 8

A/B

B/A ,

, . , . , , . o je B , \ B CB B : CAB = A \ B = {x | x A B}. x

: A= {1, 2, 3, 4, 5, 6, 7, 8, 9} B= {1, 3, 5, 7, 9} CAB = {2, 4, 6, 8} 1.3.4. ( ) , . . , ( ), 29

, , , . . {, b}, (, b). {, b} = {b, }, : (, b) (b, ). , . , . (a, b) = (x, y) (a = x) (b = y) {, b} : a. .b (, b) : a. .b B, x B, , B:

A B = { ( a, b); a A b B} 1: A = { 1, 2,3} B = { x, y}

A B = { (1, x), (2, x), (3, x), (1, y), (2, y), (3, y)}

B A = { ( x, 1), ( x, 2), ( x, 3), ( y, 1), ( y, 2), ( y, 3) } A B B A , .

30

A A A2 ; R R = R 2 , .

R R = { ( x, y); x R y R . } . 2: = {1, 2, 3} B = {4,