Relation

1. Relation

2. Indonesia consist of land and water. Indonesia have so many big island such as Sumatra, Java, and Kalimantan. On each island consist of so many town such as Medan, Palembang, Jakarta, Surabaya and Pontianak. Medan in Sumatera island. Now, discuss with your friend with your friends on group about another country. How to show the location of the town? Set A={Sumatra, Java, Kalimantan} Set B={Medan, Palembang, Jakarta, Surabaya, Pontianak}

3. Medan and Palembang located on Sumatra • Jakarta and Surabaya located on Java • Pontianak located on Kalimantan Located on is relation which joint each city and the island

4. Definition of relation • A relation from Set A to Set B is a regulation which connects the members of Set A to the members of Set B.

5. How to show relation? • By using arrow diagram • By using ordered pairs • By using Cartesian coordinate

6. Showing relation by using arrow diagram • Create two ovals with the elements of first set on the left and the elements of second set on the right. • Elements are not repeated. (when you find one element raise more than 1 time. Write only once) • Connect elements of first set with the corresponding elements in the second set by drawing an arrow.

7. Example located on Medan Palembang Sumatra Jakarta Java Surabaya Borneo Pontianak

8. Showing Relation by Using Ordered Pairs • Order = urutan • Pairs = pasangan • Ordered pairs  arrange the pairs well. • Put each element which relate to another element on bracket as a pairs, and all pairs put in one parenthesis

9. Example • {(Medan, Sumatra),(Palembang, Sumatra),(Jakarta, Java),(Surabaya, Java),(Pontianak, Kalimantan)}

10. Showing Relation by Using Cartesian Coordinate • All elements of first set put on x-axis • All elements of first set put on y-axis

11. If given the number of two different sets, how many it possible relation? Number of possible relation from two different sets called Cartesian product Suppose: Number of elements of set A is a Number of elements of set B is b • Number of Cartesian product can be determine by axb

12. Example • A={1,2,3,4,5} • B={1,2,3} • Number of possible relation from set A into set B? • n(A)=number of element of set A • n(A)=5 • n(B)=3 • Number of possible relation from set A into set B = n(A)xn(B)=5x3=15

13. Exercise 1. Identify the elements of set A and set B Name the relation Show the relation by using ordered pairs Show the relation by using Cartesian coordinate

14. 2. Identify the elements of set P and set Q Name the relation Show the relation by using ordered pairs Show the relation by using arrow diagram

15. Name the Relation • Choose one name for that relation, and check for all connection you find. • If all connection fulfill that name, so name you choose is the correct relation

16. Important Word • Domain = daerahasal • Co-domain = daerahkawan • Range = daerahhasil • Function = fungsi = pemetaan

17. Indicator of Domain • Arrow diagram : all elements on the first oval(set) • Ordered pairs : all first elements on each pairs • Cartesian coordinate : all elements on x-axis • If the elements of each set raised more than 1 time, you only write that element once.

18. Indicator Co-domain • Arrow diagram : all elements on the second set (oval) • Ordered pairs : all second element on each pairs • Cartesian coordinate : all elements on y-axis

19. Indicator Range • Arrow diagram : all elements of co-domain which have relation to domain • Ordered pairs : all second element of each pairs • Cartesian coordinate : all element of co-domain which have relation to domain

20. Example Domain = {salt, sugar, vinegar, pepper} Co-domain = {sour, salty, bitter, sweet, hot} Range = { sour, salty, sweet, hot}

21. Type of Relation • One to one relation • One to many relation • Many to one relation • Many to many relation

22. How to identify which Graph is function

23. Graphs of a Function Vertical Line Test: If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function.

24. Yes D: all reals R: all reals Yes D: all reals R: y ≥ -6 x x y y Does the graph represent a function? Name the domain and range.

25. No D: x ≥ 1/2 R: all reals No D: all reals R: all reals x x y y Does the graph represent a function? Name the domain and range.

26. Yes D: all reals R: y ≥ -6 No D: x = 2 R: all reals x x y y Does the graph represent a function? Name the domain and range.

27. The number of Function of two Sets If the number of element sets A is n(A) = a and the number of element sets B is n(B) = b, so: • The number of the possible function of sets A to B = (b)a • The number of the possible function of sets B to A = (a)b

28. Example Given A = {4,5,6} and B = {3,5} Determine the number of the possible function of: • A to B • B to A

29. Answer A = {4,5,6} n(A) = a = 3 B = {3,5} n(B) = b = 2 So: • The number of function of A to B = (b)a = 23 = 8 • The number of function of B to A = (a)b = 32 = 9

30. Correspondence One to One • Definition : Correspondence one to one of sets A and sets B is the relationship which relates every member of set A to exactly one member of set B and relates every member of set B to exactly one member of set A. The number of elements sets A and sets B are equal

31. The number of correspondence one to one If n(A)=n(B)=n,the number of possible correspondence one to one A and B is; n x (n-1) x (n-2) x….3x2x1

32. example How many the number of corraspondence one to one between sets P and Q, if P = {a, b, c, d} and Q = {3, 5, 7, 9}

33. answer n(P) = 4 and n(Q) = 4 The number of correspondence one to one P and Q = 4 x 3 x 2 x 1 = 24 ways

34. Number of one to one correspondence is 720.What is the numbers of each elements of each set which construct that correspondence? n=6

35. Function Notation • When we know that a relation is a function, the “y” in the equation can be replaced with f(x). • f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’. • The ‘f’ names the function, the ‘x’ tells the variable that is being used.

36. Symbol the function If we have function g and map x into x2-2, we symbol: • g : x  x2-2 or • g(x) = x2-2 or • y = x2-2

37. Value of a Function Since the equation y = x - 2 represents a function, we can also write it as f(x) = x - 2. Find f(4): f(4) = 4 - 2 f(4) = 2

38. Value of a Function If g(s) = 2s + 3, find g(-2). g(-2) = 2(-2) + 3 =-4 + 3 = -1 g(-2) = -1

39. Value of a Function If h(x) = x2 - x + 7, find h(2c). h(2c) = (2c)2 – (2c) + 7 = 4c2 - 2c + 7

40. Value of a Function If f(k) = k2 - 3, find f(a - 1) f(a - 1)=(a - 1)2 - 3 (Remember FOIL?!) =(a-1)(a-1) - 3 = a2 - a - a + 1 - 3 = a2 - 2a - 2