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Relations And Functions

Question
CBSEENMA12032523
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Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as

straight a space asterisk times space straight b space equals space open curly brackets table attributes columnalign left end attributes row cell straight a plus straight b space space space space space space space space space space space space space space space space if space straight a plus straight b less than 6 end cell row cell straight a plus straight b minus 6 space space space space space space space space space space space space if space straight a plus straight b greater or equal than 6 end cell end table close

Show that zero is the identity for this operation and each element a of the set is invertible with 6 – a being the inverse of a.

Solution

We construct the composition table as follows :
Table

*

0

1

2

3

4

5

0

0

1

2

3

4

5

1

1

2

3

4

5

0

2

2

3

4

5

0

1

3

3

4

5

0

1

2

4

4

5

0

1

2

3

5

5

0

1

2

3

4

From this table, it is clear that
0 * 0 = 0, 1 * 0 = 0 * 1 = 1, 2 * 0 = 0 * 2 = 2, 3 * 0 = 0 * 3 = 3,
4 * 0 = 0 * 4 = 4 and 0 * 5 = 5 * 0 = 5.
0 is the identity element.
Also for each a ≠ 0 in {0, 1, 2, 3, 4, 5 },
6 – a ∈ {0, 1,2, 3, 4, 5} and a * (6 – a) = a + (6 – a) – 6 = 0.
∴ 6 – a is inverse of a for each a ≠ 0 in the set {0, 1, 2, 3, 4, 5} Also 0 * 0 = 0, i.e., 0 is inverse of itself.

Some More Questions From Relations and Functions Chapter

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.

 Determine whether each of the following relations are reflexive, symmetric and transitive :

(i) Relation R in the set A = {1, 2, 3,....., 13, 14} defined as

R = {(x, y) : 3 x – y = 0}

(ii) Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x,y) : y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x,y) : x – y is an integer}

(v) Relation R in the set A of human beings in a town at a particular time given by
(a)    R = {(x, y) : x and y work at the same place}
(b)    R = {(x,y) : x and y live in the same locality}
(c)    R = {(x, y) : x is exactly 7 cm taller than y}
(d)    R = {(x, y) : x is wife of y}
(e)    R = {(x,y) : x is father of y}

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

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