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Relations and Functions
Question
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Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.

∧

1

2

3

4

5

1

1

1

1

1

1

2

1

2

1

2

1

3

1

1

3

1

1

4

1

2

1

4

1

5

1

1

1

1

5

(i)     Compute (2 * 3) * 4 and 2 * (3 * 4)
(ii)    Is * commutative ?
(iii)    Compute (2 * 3) * (4 * 5).

Answer

Let    A = {1,2, 3,4, 5}
Multiplication table is given as follows :
Table

*

1

2

3

4

5

1

1

1

1

1

1

2

1

2

1

2

1

3

1

1

3

1

1

4

1

2

1

4

1

5

1

1

1

1

5

(i) (2 *3) *4 = 1 *4 = 1
2* (3 *4) = 2* 1 = 1
(ii) Since the multiplication table is symmetrical about the diagonal starting at the upper left corner and ending at the lower right corner.
* is commutative.
(iii) 2 * 3 = 1, 4 * 5 = 1
∴ (2 * 3) * (4 * 5) = 1 * 1 = 1

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Some More Questions From Relations and Functions Chapter

If a matrix has 24 elements, what are the possible orders it can have ? Wh'at. if it has 13 elements ?

lf a matrix has 18 elements, what are the possible orders it can have ? What, if it has 5 elements?

If a matrix A has 12 elements, what arc the possible orders it can have 7 What if it has 7 elements ?

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.