To identify a relation and a function.
Hardboard, battery,
electric bulbs of two different colors, testing screws, tester, electrical
wires and switches.
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 = Ñ„
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.
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.
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.
Relation is the subset of the Cartesian product and a function is a
relation in which no two ordered pair have same first element.
With the help of this activity we can explain a relation, function and
the concept of one-one and onto functions.
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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