Mathematics Lab Activity-4 Class XI

Mathematics Laboratory Activities on Relaton & Functons for class XI students Non-Medical. These activities are strictly according to the CBSE syllabus

Chapter - 2 | relations & functions

Activity - 4

Objective

To identify a relation and a function.

Material Required

Hardboard, battery, electric bulbs of two different colors, testing screws, tester, electrical wires and switches.  

Theory

1) Cartesian Product: If A and B are two empty sets, then their Cartesian product A x B is the set of all ordered pairs of elements from A to B  i.e.

A x B ={(a, b) : a ∈ A, b ∈ B}

If there are p elements in set A and q elements in set B, then there will be pq elements in A x B. i.e. if n(A) = p and n(B) = q, then n(A x B) = pq

If either A or B is the null set , then A x B will also be null or empty set.   i. e. A x B = ф

2. Ordered pair of elements: An ordered pair of elements taken from two sets A and B is a pair of elements written in small brackets and grouped together in a particular order. i.e. (a, b), a ∈ A and b ∈ B.

3. Equal Ordered Pairs : Two ordered pairs are equal if and only if the corresponding first elements are equal and the second elements are also equal.

4. In general :  A ✕ B ≠ B ✕ A

5. Relation : A relation R from a non-empty set A to a non- empty set B is a subset of the Cartesian product A ✕ B . The subset is derived by describing a relation between the first element and second element of the ordered pairs in A ✕ B .

6. Domain of the relation: The set of all the first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.

7. Range of the relation : The set of all second elements in a relation R from a set A to a set B is called the range of the relation R.

8. Co-domain of the relation: The whole set B is called the co-domain of the relation R.

Note: Range is the subset of the co-domain i.e.  Range ⊂ Co-domain

9. Function: A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.

Note : A relation is said to be a function If no two ordered pairs have same first element. A function from A to B is denoted by  f : A → B

10. Real function and Real valued function: A function whose domain is either R or a subset of R then it is called a real  function. A function whose range is either R or a subset of R then it is called a real valued function.

Procedure

 1. Take a white paper and paste it on a piece of hardboard of suitable size.

2. Drill eight holes on the left side of the board in a column.

3. Mark these holes as A, B, C, D, E, F, G, H as shown in figure 5.1.

4. Drill seven holes on the right side of the board in a column.

5. Mark these holes as P, Q, R, S, T, U, V as shown in figure 5.1.

6. Fix bulbs of one color in the holes A, B, C, D, E, F, G, H.

7. Fix bulbs of another color in the wholes P, Q, R, S, T, U and V.

8. Fix testing screws at the bottom of the board and mark them as 1, 2, 3, 4, 5, 6, 7 and 8.

9. Make the electric circuit in such a way that a pair of corresponding bulbs, one from each column glow simultaneously.

10. These pair of bulbs will give ordered pairs, which will constitute a relation.

11. The bulbs A, B, C, D, E, F, G, H along the left column represent domain and the bulbs along the right column at P, Q, R, S, T, U and V represent co-domain.

12. Obtain different ordered pairs by using both sides of bulbs. In fig. 5.1 different ordered pairs are : (A, P), (A, R), (B, R), (C, Q), (D, T), (E, Q), (E, S), (F, U), (G, T), (H, U)

13. Similarly Make different sets of ordered pairs.

Figure 5.1

Observations

1. In Fig. 5.1, the ordered pairs are (A, P), (A, R), (B, R), (C, Q), (D, T), (E, Q), (E, S), (F, U), (G, T), (H, U).

2. These ordered pairs constitute a relation.

3. The ordered pairs (A, P), (B, R), (C, Q), (D, T), (E, Q), (F, U), (G, T), (H, U) constitute a relation which is a function.

4. The ordered pair (A, P), (A, R), (B, R), (C, Q), (D, T), (E, Q), (E, S) constitute a relation which is not a function.

Result

Relation is the subset of the Cartesian product and a function is a relation in which no two ordered pair have same first element.

applications

With the help of this activity we can explain a relation, function and the concept of one-one and onto functions.

VIVA – VOICE

Q. 1. What is an ordered pair of elements?

Ans. An ordered pair of elements taken from two sets A and B is a pair of elements written in small brackets and grouped together in a particular order.

Q. 2. What are equal ordered pair?

Ans. Two ordered pairs are said to be equal if their corresponding first elements and second elements are equal.

Q. 3. Define relation in a set theory?

Ans. A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product  A x B.

Q. 4. What do you mean by domain of a relation?

Ans. Set of  first elements of all the ordered pairs of a relation is called the domain of the relation?

Q. 5. What is range of a relation?

Ans. Set of  second elements of all the ordered pairs of a relation is called the range of the relation?

Q. 6. Define function in a set theory.

Ans. A relation f from set A to set B is said to be a function if different elements in set A has different image in set B .

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