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Relations

1. 1. M.Yafie Setyo W. 7.10/13
2. 2. contens Understanding relations Function of mapping One-to-one correspondence Function formula Graph of a functions
3. 3. Understanding relations Explanation Expressing relations
4. 4. explanation Ria and Reni like hp Reni and Revi likes samsung Rian like lenovoif a={Ria, Reni, Revi, Rian} and b={hp, samsung, lenovo} a relation among the element of the set b can be build. Figure 1.0 Ria Hp relation Reni Samsung Revi Lenovo Rian The relation from set a and set b visualized in figure 1.0 is called a favor relation.
5. 5. Expressing relations Arrow diagram Cartesian diagram Sets of ordered pairs
6. 6. Arrow diagram example:Build three types of relation from the set P={2,3,5} to the set Q={2,4,6} and express them using arrow diagram.
7. 7.  A. Is less than .2 .2 .3 .4 .5 .6 B. Is greater than .2 .2 .3 .4 .5 .6 C. Is a factor of .2 .2 .3 .4 .5 .6
8. 8. From the example,What is arrowdiagram????? A relation is represented by Arrow. Is less than .2 .2 .3 .4 .5 .6
9. 9. Cartesian diagram The relation among the element of two sets A of B can be expressed by a cartesian diagram in which the element of the set A acting as the first set lie on horizontal axis and the element of the set B acting as the second set lie on vertical axis . Figure 1.1 Cartesian diagram A. 2.
10. 10. Sets of the ordered pairs A relation among the element of two sets K and L can be expressed as an ordered pairs(x,y) in which x e K and y e L is Paired
11. 11. Function of mapping Understanding function of mappings Expressing function or mappings Number of possible ways of mapping between two sets
12. 12. Understanding function of mappings Let A and B be any two non-empty sets. Let A = {p, q, r} and B = {a, b, c, d}. Suppose by some rule or other, we assign to each element of A a ‘unique’ element of B. Let p be associated to a, q be associated to b, r be associated to c etc. The set {(p, a), (q, b), (r, c) } is called a function from set A to set B. If we denote this set by ‘f’ then we write f : A ® B which is read as "f " is a function of A to B or ‘f’ is a mapping from A to B.
13. 13. example Determine whether or not the following arrow diagram express mapping. B B A B A A .a .u .u .u .a .a .b .v .v .v .b .b .c .w .w .w .c .c .x .x .x 1 2 3
14. 14. Answer is……. Figure 1. does not express a mapping since there exits an element of A,namely b,which is paired with more than one element of B. Figure 2. does express a mapping since every element of A paired with exactly one element of B. Figure 3. does not express a mapping since there exits an element of A,namely b,which is paired with no element of B.
15. 15. A B .a .1 Image (map) of a .b .2 .c .3 range .d .4domain codomain
16. 16. explanation A={a,b,c,d} is called domain B={1,2,3,4} is called codomain {2,3,4} which called range , is set of the elements of Q are paired with the elements of P. That element a is paired with 2 can be denoted by a- 2,which is read “ a is mapped to 2" in the form of a-2.2 called the image or map from a.
17. 17. Expressing function or mappings In previous section we have stated that function is a special relation.therefore,a function can be expressed by means of any of the following three expressions. 1. arrow diagram 2.cartesian diagram 3.Sets of ordered pairs
18. 18. Number of possible ways ofmapping between two sets 1 possible (A={a,b} to B={p}) .a .p .b 2.possible (A={a} to B={p,q}) .a .a .p .p .q .q
19. 19.  8 possible (A={a,b,c} to B={p,q} ).a .a .p .p .a.b .b .p .q .q .b.c .c .q .c.a .a .a .p .p .p.b .b .b .q .q .q.c .c .c.a .a .p .p.b .b .q .q.c .c
20. 20.  9 possible (A={a,b} B={p,q,r}) .a .p .a .a .b .p .q .b .b .p .q .q .r .r .r.a.b .p .a .q .b .p .a .q .b .p .r .r .q .r.a .a .a.b .p .p .p .b .b .q .q .q .r .r .r
21. 21. One to one correspondence P P Q QIndonesia. Jakarta. Jakarta. Indonesia.Malaysia. Kuala lumpur. Kuala Malaysia.Thailand. Bangkok. lumpur. Thailand.Singapore. Singapore. Bangkok. Singapore.philipine Manila. Singapore. philipine Manila.
22. 22.  In above figure every country is paired with exactly one capital,while in above figure every capital is paired with exactly one country.There comes into play so called reciprocal mapping between the set P and Q,hence a One to one correspondence.  Similarly,every country has only one national anthem,hence a One to one correspondence between the set of countries and the set of national anthems . Since a two sets having a One to one correspondence can be connected using bidirectional arrows as shown below. National Indonesia raya Negaraku God save Kimigayo anthem The queen Nations indonesia Malaysia great Japan britain
23. 23. Functions Formula Formulating functions Independent variable and dependent Variable
24. 24. Formulating functions A mapping by a function f that maps every element x of a set A to an element y of a set can be denoted by. f:x y The notation f : x y is read:function f maps x to y. here ,y is called the image (map) of x under f Figure 2.0 shows a functions f mapping A to B. if x an element of the domain of A,then the image of x under f is denoted by f(x),and is read a function of x. Image 2.1 describes the mapping f:x x+2.Since the image of x under f can be denoted by f(x),we can express the mapping as f(x) =x+2. The form f(x)=x+2 is called function formula.
25. 25. Figurex f(x) 2.0x x+2 Figure 2.1
26. 26. example Determine the function formula for each the following functions. Functions f:x 4x+1 Answer; f:x 4x+1,is formulated as f(x)=4x+1
27. 27. Independent variable anddependent Variable Dependent:A variable that depends on one or more other variables. For equations such as y = 3x – 2, the dependent variable is y. The value of y depends on the value chosen for x. Usually the dependent variable is isolated on one side of an equation. Formally, a dependent variable is a variable in an expression, equation, or function that has its value determined by the choice of value(s) of other variable(s).
28. 28.  Independent: A variable in an equation that may have its value freely chosen without considering values of any other variable. For equations such as y = 3x – 2, the independent variable is x. The variable y is not independent since it depends on the number chosen for x. Formally, an independent variable is a variable which can be assigned any permissible value without any restriction imposed by any other variable.
29. 29. Graph of a function The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). So, the graph of a function if a special case of the graph of an equation.
30. 30. example Let f(x) = x2 - 3. Recall that when we introduced graphs of equations we noted that if we can solve the equation for y, then it is easy to find points that are on the graph. We simply choose a number for x, then compute the corresponding value of y. Graphs of functions are graphs of equations that have been solved for y! The graph of f(x) in this example is the graph of y = x2 - 3. It is easy to generate points on the graph. Choose a value for the first coordinate, then evaluate f at that number to find the second coordinate. The following table shows several values for x and the function f evaluated at those numbers. x -2 -1 0 1 2 F(x) 1 -2 -3 -2 1 Each column of numbers in the table holds the coordinates of a point on the graph of f.
31. 31. Example 2 Functions of one variable The graph of the function. Is {(1,a), (2,d), (3,c)}. The graph of the cubic polynomial on the real line is {(x, x3-9x) : x is a real number}.If this set is plotted on a Cartesian plane, the result is a curve (see figure).
32. 32. Exercise1. A B .1 .2 .2 .4 .3 .6The arrow diagram shown above expresses a……relationsA.is more thanB.is less thanC.is the square ofD.is a factor of
33. 33.  2. P= {3,4,5} and Q= {1,2,3,4,5,6,7}.  The set of ordered pairs which expresses the relation “is two more than” from the set of P to the set of Q is.... A.{(3,2),(4,2),(5,2)} B.{(3,4),(4,5),(5,6)} C.{(3, 1),(4,2),(5,3)} D.{(3, 5),(4,6),(5,7)} 3. i ii iii iv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Among the arrow diagram shown a.(i) and (ii) b.(i) and (iii) Above,those which express c.(ii) and (iii) Functions are… d.(ii) and (iv)
34. 34.  4. a. .p The range of the .b .q function expressed by c. .r the arrow diagram shown is…… a. {p,r} b. {a,b,c} c. {p,q,r} d. {a,b,c,p,q} 5.Among the following sets of ordered (i) {(a,1),(a,2),(a,3),(a,4)} Those which have a one (ii) {(a,2),(a,2),(a,2),(a,2)} to one correspondence are…. (iii) {(a,1),(b,2),(c,1),(d,2)} a.(i) b. (ii) (iv) {(a,1),(b,2),(c,3),(d,4)} c. (iii) d. (iv)
35. 35. 1.build a arrow diagram expressing the relation “is a factor” from the set K=(0,1,2) to the setL=(4,5,6)2.build an arrow diagram for each possible specification of one to one correspondence between the set P={1,2} and the set Q={a,b}3.A={letter forming word”pandai”} B={letter forming word”babat””}Determine the number the number of possible ways of mappings.a.From A to Bb.From B to A4.A relation on the set A={0,2,4,6,8} is expressed by x is evenly divided by y where x,y A, express the relation as a set of ordered pairs (x,y)!5.Let the function f:x 2x+1Build a table for the function f from{-3,-2,-1,0,1,2,3,4}to the set of integers!
36. 36. Thank you