1. M.Yafie Setyo W.7.10/13

2. contens Understanding relations Function of mapping One-to-one correspondence Function formula Graph of a functions 3. Understanding relations Explanation Expressing relations 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 ReniSamsung ReviLenovo RianThe relation from set a and set b visualized in figure 1.0 is called a favor relation. 5. Expressing relations Arrow diagram Cartesian diagram Sets of ordered pairs 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. 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. From the example,What is arrowdiagram????? A relation is represented by Arrow. Is less than.2.2.3.4.5.6 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. 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. Function of mapping Understanding function of mappings Expressing function or mappings Number of possible ways of mapping between two sets 12. Understanding function of mappingsLet 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. example Determine whether or not the following arrow diagram express mapping.BBA BAA .a .u.u.u .a.a .b .v.v.v .b.b .c .w.w.w .c.c.x.x.x1 2 3 14. Answer is. Figure 1. does not express a mapping since there exitsan element of A,namely b,which is paired with morethan one element of B. Figure 2. does express a mapping since every elementof A paired with exactly one element of B. Figure 3. does not express a mapping since there exitsan element of A,namely b,which is paired with noelement of B. 15. AB.a .1 Image (map) of a.b .2.c .3 range.d .4domain codomain 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 Qare 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-2.2called the image or map from a. 17. Expressing function or mappings In previous section we have stated that function is aspecial relation.therefore,a function can be expressedby means of any of the following three expressions. 1. arrow diagram 2.cartesian diagram 3.Sets of ordered pairs 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. 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. 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. One to one correspondenceP P Q QIndonesia. Jakarta.Jakarta. Indonesia.Malaysia.Kuala lumpur. KualaMalaysia.Thailand.Bangkok.lumpur.Thailand.Singapore. Singapore.Bangkok. Singapore.philipineManila. Singapore. philipine Manila. 22. In above figure every country is paired with exactlyone capital,while in above figure every capital is pairedwith exactly one country.There comes into play socalled reciprocal mapping between the set P andQ,hence a One to one correspondence. Similarly,every country has only one nationalanthem,hence a One to one correspondence betweenthe set of countries and the set of national anthems .Since a two sets having a One to one correspondence canbe connected using bidirectional arrows as shownbelow. National Indonesia raya Negaraku God save Kimigayo anthem The queenNations indonesiaMalaysia greatJapan britain 23. Functions Formula Formulating functions Independent variable and dependent Variable 24. Formulating functions A mapping by a function f that maps every element x of aset A to an element y of a set can be denoted by.f:x y The notation f : xy 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 anelement of the domain of A,then the image of x under f isdenoted by f(x),and is read a function of x. Image 2.1 describes the mapping f:x x+2.Since theimage of x under f can be denoted by f(x),we can expressthe mapping as f(x) =x+2.The form f(x)=x+2 is called function formula. 25. Figurex f(x) 2.0x x+2Figure 2.1 26. example Determine the function formula for each the followingfunctions. Functions f:x 4x+1 Answer; f:x 4x+1,is formulated as f(x)=4x+1 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. Independent: A variable in an equation that may have itsvalue freely chosen without considering values of any othervariable. For equations such as y = 3x 2, the independentvariable is x. The variable y is not independent since itdepends on the number chosen for x. Formally, an independent variable is a variable which canbe assigned any permissible value without any restrictionimposed by any other variable. 29. Graph of a function The graph of a function f is the set of all points in theplane of the form (x, f(x)). We could also define thegraph 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 ofan equation. 30. example Let f(x) = x2 - 3. Recall that when we introduced graphs of equations wenoted that if we can solve the equation for y, then it is easy tofind points that are on the graph. We simply choose anumber for x, then compute the corresponding value of y.Graphs of functions are graphs of equations that have beensolved for y! The graph of f(x) in this example is the graph of y = x2 - 3. It iseasy to generate points on the graph. Choose a value for thefirst coordinate, then evaluate f at that number to find thesecond coordinate. The following table shows several valuesfor x and the function f evaluated at those numbers.x -2-10 12F(x)1 -2-3-2 1 Each column of numbers in the table holds the coordinatesof a point on the graph of f. 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 aCartesian plane, the result is a curve (see figure). 32. Exercise1.AB .1 .2 .2 .4 .3 .6The arrow diagram shown above expresses arelationsA.is more thanB.is less thanC.is the square ofD.is a factor of 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 twomore 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 iiiiv . ... . . .. . ... . . .. . ... . . .. . ... . . .. Among the arrow diagram showna.(i) and (ii)b.(i) and (iii) Above,those which expressc.(ii) and (iii) Functions are d.(ii) and (iv) 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 correspondenceare. (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. 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 wordpandai}B={letter forming wordbabat}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. Thank you