DIPLOMSKI - Uloga i Znacaj Skupova u Pocetnoj Nastavi Matematike

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<p>................................................................................................................................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</p> <p>2</p> <p>3.2. .....................................38 3.3. ........................................................................39 ................................................................................................................41 .............................................................................................................43 .....................................................................................................................44 1. .............................44 2. .............................................46</p> <p>3</p> <p> , , . UNESCO, 1956. . , , , , . , , , , . . , : episteme mathema. , , , . mathema mathematika ( ), . , , , , , . , , , , , . . . .</p> <p>4</p> <p> , . XIX , . , . , ( ) . , , , . . , . . , , . , , . ( ). ( ) , . , , , . , . , , , , : , . , 5</p> <p>, , . . ( , ...) ( ) . . , . , . , . , , , . ; ; ( ). , , : , . . , . : ? . . , , , 6</p> <p> . , . , , , . , . , . , .</p> <p>7</p> <p>1. </p> <p>1.1. . . , , . , , , , , . , , . , : ( ), ( ), ( ) , , : , , , , . , . , . , , , ., - . . . . . : , , . . 8</p> <p> . , . , (1848 1918), , , (1871 1956). ( ), . , : - , . - . - . . . . , , , . , . , 100 . , . , , , , , 1881. . , , , XVIII . , ; 9</p> <p> . , , . :</p> <p> :</p> <p>*** , , , , ; je je . Ha , (3) (3) , je ( ). , , .</p> <p>10</p> <p>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&lt; 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</p> <p>19</p> <p> . . 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. () , .:</p> <p>* * * *</p> <p>* *</p> <p>* * *</p> <p>* *</p> <p> 1: A={a, e, , o, u }, |A| = 5 k(A) = 5 2: . . .</p> <p>20</p> <p> , . (3).</p> <p>*** . , , . . , . , . , . . : , , . .</p> <p> . , . , , </p> <p>21</p> <p>, . , , ( ). .</p> <p> . , , , , , ... , , . : , . , , . , : , , . , : , . , : , , . , : , . , , </p> <p>22</p> <p> , . . , , , . . . , . . , . , . : , , , , . , : ? ( ). ? ? ? , , , . . , , (, ...). , , . , .</p> <p>23</p> <p>1.2.4. . . : </p> <p> . . , N= {1, 2, 3, ...}, N0= {0, 1, 2, 3, ...}.</p> <p> ( , ). = {..., -2, -1, 0, 1, 2, ...}.</p> <p> (. ). ={ : a,b b 0}. ,</p> <p>. .</p> <p> . (. , 2).</p> <p> . , , </p> <p> . , . () ( ). , , </p> <p> ( ), </p> <p>P, N, Z, Q, C.</p> <p> , , 3, 4, . R </p> <p>24</p> <p>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</p> <p>A</p> <p>B</p> <p>C</p> <p>=</p> <p>C=AB *** . , , . , . , </p> <p>25</p> <p> B. B. C. C B, . C = B : UB B</p> <p> , . . . , . . . , , . , , . , . 1: A={1, 2, 3, 4} B={3, 4, 5, 6, 7} AUB={1, 2, 3, 4, 5, 6, 7}</p> <p>26</p> <p> 2: A = {, , } B = { , , , } AUB = {, , , , , , } BUA = {, , , ,, , }</p> <p> A B</p> <p>AUB</p> <p>1.3.2. B. B. A B = {x | x A x } :</p> <p>: 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</p> <p> A B = B A. B , . B = . : </p> <p>AB=BA A B A AA=A A=</p> <p>: : 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 = </p> <p>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}</p> <p>28</p> <p>A B 3 6 4 5 7 8</p> <p>A/B</p> <p>B/A , </p> <p> , . , . , , . o je B , \ B CB B : CAB = A \ B = {x | x A B}. x</p> <p>: 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</p> <p> , , , . . {, b}, (, b). {, b} = {b, }, : (, b) (b, ). , . , . (a, b) = (x, y) (a = x) (b = y) {, b} : a. .b (, b) : a. .b B, x B, , B:</p> <p>A B = { ( a, b); a A b B} 1: A = { 1, 2,3} B = { x, y}</p> <p>A B = { (1, x), (2, x), (3, x), (1, y), (2, y), (3, y)}</p> <p>B A = { ( x, 1), ( x, 2), ( x, 3), ( y, 1), ( y, 2), ( y, 3) } A B B A , .</p> <p>30</p> <p> A A A2 ; R R = R 2 , .</p> <p>R R = { ( x, y); x R y R . } . 2: = {1, 2, 3} B = {4, 5} AxB = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}</p> <p>1 2 3</p> <p>4 5</p> <p>3 2 1 1 2 3</p> <p>4) (3 (3 4) (2 (2 4) (1 (1 4 5</p> <p>5) 5) 5)</p> <p> B , x B = A2 A = {a,b} A2 = A x A = {(a, a), (a, b), (b, a), (b, b)}</p> <p>a b</p> <p>a b</p> <p>31</p> <p>32</p> <p>2. 2.1. . . , . : ? , . : . 2.2. , , , . 2.3. , . . 2.4. : 1. .</p> <p>33</p> <p>2. . 3. . 2.5. : 1. . 2. . 3. . 2.6. , . , , , , . . . . : 1. : - , - .</p> <p>34</p> <p>2. : - , - , - . 2.7. : . . (, ) . . , , , , . 2.8 , . , , . , . ( 1) . : 1. ; 2. ; 3. .</p> <p>35</p> <p> . , . 2.9. , , . , , , , . ( ., 2002: 294). : , . 1: 2.10. 2009. . . 36 50 20 30 100 % 50 % 20 % 30 % 100,00</p> <p> 100 . , . 2.11. . , : 1. ; 2. ; 3. . :a) :</p> <p>(0 t)</p> <p>=</p> <p>t</p> <p>2</p> <p> : 0 t</p> <p> () </p> <p>) = ______ N+ </p> <p> : </p> <p>37</p> <p>3. 3.1. . 1 5. . : 1 ; 2 ; 3 ; 4 ; 5 . : 2: 0 6 22 49 23 100 0% 6% 22 % 49 % 23 % 100,00</p> <p> 22% , 49% , 23% . , 6% . . : 3: ( )</p> <p>38</p> <p> 0 t 0 - t (0 t)2 (0 t)2 t</p> <p> 6 5 1 1 1/5=0,2</p> <p> 22 15 7 49 49/15=3,2</p> <p> 72 80 8 64 64/80=0,8 df = 2 p = 0,01</p> <p> 100 100 4,2</p> <p> = 4,2</p> <p>C = 0,205</p> <p> = 4,2 (5,991) (df=2) 0,01 , , .</p> <p>100 80 60 40 20 0 </p> <p> 1: </p> <p>39</p> <p>3.2. . . (, , ), . 100 , 83% , 16% , , 1% . 4. 4: 0 83 t 80 0 - t 3 2 (0 t) 9 (0 t)2 9/80=0,112 t = 3,372</p> <p> 1 5 -4 16 16/5=3,2 C = 0,033</p> <p> 16 15 1 1 1/15=0,06 df = 2 p = 0,01</p> <p> 100 100 3,372</p> <p> = 3,372 (5,991) (df=2) 0,01 . , .</p> <p>40</p> <p>100 90 80 70 60 50 40 30 20 10 0</p> <p> (84) (1) (16) </p> <p> 2: </p> <p>3.3. , , . , , . 100 , 68% , 20% , 12% . . 5 . 3.</p> <p>41</p> <p> 5: 0 12 t 5 0 - t 7 (0 t)2 49 (0 t)2 49/5=9,8 t = 13,266 20 15 5 25 25/15=1,666 C = 0,342 68 80 -12 144 144/80=1,8 df = 2 100 100 13,266</p> <p>p = 0,01</p> <p> = 13,266 (5,991 9,210) df=2 0,05 0,01, . , .</p> <p>100 80 60 40 20 0 </p> <p> 3: . . </p> <p>42</p> <p> , , , , . , , . , , . , , . , . , , . . , . , , . , , . , : ? , . je , a je , . . , , 43</p> <p>. . . . : ( ), ( ) , . . , , . , . a je . . , . , , - . , . . 68% , . (, , ...), . . , , , . , - , , , , , 44</p> <p>1. 2. 3. 4. 5. 6. 7. 8. 9.</p> <p>, . , . (2003): . : , . (2000): . : , . . (1960): . : , . (1971): , , . : , . (1991): . : , . (2003): . : , . (1996): , . : , . . (2007): VI . : , . , . (2008): I . : </p> <p>10. http://www.fesb.hr/~borka/files/Diskretna1-05.pdf 11. http://www.matematiranje.com 12. http://www.chemistrydaily.com/chemistry/Venn_diagram 13. http://www.fpmoz.ba/matematika/UMMostar03-skupovi.pdf</p> <p>45</p> <p> 1. : . . . . 1. 2. 3. 4. 5. 6. : : : : 5 7. 4 3 2 1 . ) ) ) ) </p> <p> , </p> <p>8.</p> <p> ?</p> <p>) </p> <p>) </p> <p>9.</p> <p> ? ) ) ) </p> <p>10 .</p> <p> :</p> <p>46</p> <p>) ) ) 11 . ) ) ) ) 12 . , : 13 . 14 . ) ) , :</p> <p> ? , .</p> <p> ! </p> <p>47</p> <p> 2.</p> <p>1. 2. 3. 4. 5. 5.</p> <p>: : : : :</p> <p>II , (3) </p> <p> , . , , .</p> <p>5. :</p> <p>5. :</p> <p> ; , .</p> <p>6.</p> <p> , , .</p> <p>7.</p> <p> :</p> <p>, .</p> <p>8. 9.</p> <p> : :</p> <p> .</p> <p> , .</p> <p>48</p> <p>) (10 ) . , ? () ? () ? () ? ( ). ) (30 ) , , , . ? ( ). , . ? (...</p>