Discrete mathamatic by adeel
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- 3. Cartesian Products Definition : • Let A and B be two sets. The Cartesian product of A and B, denoted AxB, is the set of all ordered pairs (a,b) such that a belongs A and b belongs B. • 4/5/2015 1 3 {( , ) | and }.A B a b a A b B
- 4. Example: Finding Cartesian Products • Let A = {a, b}, B = {1, 2, 3} • Find each set. • a)AxB • b)BxB • Solution • a) {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)} • b) {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), • (3, 1), (3, 2), (3, 3)} 4/5/2015 1 4
- 5. Domain • The domain of a function is the complete set of possible values of the independent variable. • The domain is the set of all possible x- values which will make the function "work“ 4/5/2015 1 5
- 6. Range • The range of a function is the complete set of all possible resulting values of the dependent variable • The range is the resulting y-values we get after substituting all the possible x-values. 4/5/2015 1 6
- 7. A relation is a set of ordered pairs. {(2,3), (-1,5), (4,-2), (9,9), (0,6)} This is a relation The domain is the set of all x values in the relation {(2,3), (-1,5), (4,-2), (9,9), (0,6)} The range is the set of all y values in the relation {(2,3), (-1,5), (4,-2), (9,9), (0,6)} domain = {-1,0,2,4,9} range = {-6,-2,3,5,9} 4/5/2015 1 7
- 8. Domain (set of all x’s) Range (set of all y’s) 1 2 3 4 5 2 10 8 6 4 A relation assigns the x’s with y’s This relation can be written {(1,6), (2,2), (3,4), (4,8), (5,10)} 4/5/2015 1 8
- 9. One- to - many One- to - one many –to - one Types of Relations 4/5/2015 1 9
- 10. A function f from set A to set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B. Set A is the domain 1 2 3 4 5 Set B is the range 2 10 8 6 4 A function f from set A to set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B. A function f from set A to set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B. This is a function ---it meets our conditions All x’s are assigned No x has more than one y assigned 4/5/2015 1 10
- 11. 4/5/2015 1 11 A good example that you can “relate” to is students in our maths class this semester are set A. The grade they earn out of the class is set B. Each student must be assigned a grade and can only be assigned ONE grade, but more than one student can get the same grade (we hope so---we want lots of A’s). The example show on the previous screen had each student getting the same grade. That’s okay. 1 2 3 4 5 A E D C B Is the relation shown above a function? NO 2 was assigned both B and E A good example that you can “relate” to is students in our math class this semester are set A. The grade they earn out of the class is set B. Each student must be assigned a grade and can only be assigned ONE grade, but more than one student can get the same grade (we hope so---we want lots of A’s). The example shown on the previous screen had each student getting the same grade. That’s okay.
- 12. How we identify function or Not • If two or more pair have same value of X then that condition is not function. • (2,3),(2,4),(2,5),(2,1) • If all the pair same value of y then no effect on function. • (2,3),(4,3),(6,3),(0,3) 4/5/2015 1 12
- 14. • (2,3),(4,3),(6,3),(0,3) 4/5/2015 1 14 0,0 y Y’ xX’ -1-2-3-4-5-6 -1 -2 -3 -4 -5 654321 5 4 3 2 1 -6 6 6,34,32,30,3
- 18. Graph • showing a relationship (usually between two set of numbers) by means of a line, curve, or other symbols. • Typically, an independent variable is represented on the horizontal line (X-axis) and an dependent variable on the vertical line (Y-axis). • The perpendicular axis intersect at a point called origin 4/5/2015 1 18
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