Unit 1. Algebra- Relations and functions

من طرف ayman akl الخميس 09 فبراير 2012, 2:20 pm

Unit 1
Relations and functions

1 – 1): Cartesian product

Ø In each ordered pair (a , b), a is called the first projection, and b is called the second projection.

Ø Each pair is represented by one and only one point in the coordinate plane.

Ø (x , y) (a , b)

Ø (x , y) {a , b}

Ø If ( x , Y ) = (a , b), then x = a, y = b.

For example: if (x , y) = (3 , 5), then x = 3 and y = 5.

Ø X × φ = φ × X = φ, where φ is the empty set.

Example: if (x – 5 , 7) = (3 , y + 1). Find the value of x and y.

Solution:x – 5 = 3 x = 8, 7 = y + 1 y = 6.

Exercises: Find the value of a , b in each of the following cases.

1. (a , b) = (1 , 4).

2. (a , b) = (3 , 4).

3. ( , 3) = (9 , b -1).

4. (-a , b) = (-3 , 5).

5. (a – 1 , b) = (1 , 6).

6. (a + 1 , 3) = (1 , 3b – 1).

7. ( , b + 2) = (4 , 9).

8. (3 , b) = (5a – 1 , 4a).

9. (a – 2 , 3) = (5 , ).

10. (a – 2 , b + 1) = (2 , -3).

11. (6 , b – 3) = (2 – a , -1).

Cartesian product:

If X = {2 , 3}, Y = {4 , 5 , 7}, then the Cartesian product of the set X and Y which denoted by the symbol X × Y is the set of all ordered pairs in which the first projection is an element of X and the second projection is an element of Y. This Cartesian product is written as:

X × Y = {(2 , 4) , (2 , 5) , (2 , 7) , (3 , 4) , (3 , 5) , (3 , 7)}.

Notice that: Y × X = {4 , 5 , 7} × {2 , 3} = {(4 , 2) , (4 , 3) , (5 , 2) , (5 , 3) , (7 , 2) , (7 , 3)}.

There for X × Y Y × X when X Y.

Remarks:

If X and Y two are two finite and non empty sets then:

1. X × Y = {(a , b) : a X, b Y}.

2. n (X × Y) = n (Y × X ) = n (X) × n (Y), where n denoted the number of set elements.

3. If X is non empty set, then X × X = {(a , b) : a , b X} and written as and read as (X two).

4. If (s , l) X × Y, then s X and l Y.

Exercise:

1. If X = {a , b} and Y = {3 , 4 , 5}. Identify for which Cartesian product each of the following ordered belongs to.

1. (a , 4), 2. (b , 5), 3. (4 , b), 4. (5 , 5), 5. (a , b), 6. (4 , 3).

2. If n (X × X) = 9: find the elements of X such that (3 , 4), (-2 , 4) X × X.

3. If X = {1 , 3}, Y = {3 , 4 , 5} and Z = {3}. Find:

1. (X Y) × Z, 2. (Y × Z), 3. X × (Y – Z) 4. n {(X Z) × Y}.

The representation of the Cartesian product:

If X = {1 , 5} and Y = {2 , 3 , 7}, then X × Y = {(1 , 2) , (1 , 3) , (1 , 7) , (5 , 2) , (5 , 3) , (5 , 7)}. We can represent X × Y by two methods:

1^st: the arrow diagram.

In the arrow diagram an arrow drawn from each element of set X which represents the 1^st projection to each element in Y which represents the 2^nd

2^nd: Cartesian diagram.

On a perpendicular graph net, the element of set X is represented horizontally and the element the element of set se of Y represented vertically. The intersection points of horizontal and vertical lines represent the ordered pairs of the element of the Cartesian product X × Y

Example

If X = {5 , 7}, find X × X and represent it by arrows diagram.

Solution : X × X = = {(5 , 5), (5 , 7), (7 , 5), (7 , 7)}.

I. answers the following questions:

1. If X = {2 , 5}, Y = {1 , 4 , 5}. Find:

Ø X × Y and represent it by arrows diagram and Cartesian diagram.

Ø And represent it by arrows diagram and Cartesian diagram.

Ø n (X × Y).

2. If X = {2 , 1 , 0}, Draw the arrows diagram and the Cartesian diagram to X × X.

3. if X = {1 , 3}, Y = {1 , 4} and Z = {2 , 5}. Identify the Cartesian product which the following ordered pairs belongs to it.

1. (3 , 5), 2. (5 , 5), 3. (3 , 4), 4. (3 , 3), 5. (4 , 5), 6. (2 , 3).

4. If X = {7 , 9}, Y = {1 , 3 , 8}. Identify the Cartesian product which the following ordered pairs belongs to it.

1. (7 , 7) 2. (7 , 3) 3. (3 , 9) 4. (8 , Cool

.

5. X = {3 , 4}, Y = {5 , 7 , 8}. Find:

1. X × Y and represent it par arrows diagram.

2. and represent by arrows diagram.

3. Y× X and represent it by Cartesian diagram.

6. If X × Y = {(1 , 1) , (1 , 4), (1 , 5)}. Find each of the following: 1. X and Y 2. n (Y× X), 3.

7. If X × Y = {(3 , 4) , (3 , 5), (3 , Cool

}. Find each of the following:

1. X and Y 2. n (Y× X), 3.

8. If A = {1 , 2}, B = {3 , 4}, C = {1 , 3}. Find: 1. (A B) × (B C). 2. A × (B C).

9. If X = {1 , 2}, Y = {2 , 3}, Z = {1 , 3}. Find each of the following:

1. X × (Y Z). 2. (X Y) × (X Z). 3. (X – Z) × (X Y). 4. (Y – X) × Z.

10. If X = {1 , 4 , 7}, Y = {4 , 7 , 13}, Z = {1 , 7}. Find:

1. (X – Y) × Z. 2. (Z – X) × Y 3. (X Z) × Y 4. (X × Y).

11. If X = {1 , 2 , 3}, Y = {2 , 3 , 4}. Prove that: X × (X Y) = (X × X) (X × Y).

12. If n (X × Y) = 9, and n (Y) > 1. Find the element of the set X if (2 , 2), (3 , 5) X × X.

13. If n (X × Y) = 16, (2 , 3), (3 , 4), (4 , 5) Y × Y. Identify the element of the set Y.

14. If X × Y = {(K , 1) , (K , 2) , (K , 3) , (2 , 1) , (2 , 2), (2 , 3)} and X Y. find the value of K.

AYMAN AKL Maths teacher

من طرف **ahmed badr** الخميس 09 فبراير 2012, 11:07 pm

thanks my ayman . we have to get many shares from you . i think they will be so fruitful for the students and teachers at scool as well .

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Unit 1. Algebra- Relations and functions

Unit 1. Algebra- Relations and functions

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