Chapter 1 Relations and Functions Ex 1.4 Solutions
Question - 11 : - LetA=N x N and * be the binary operation on A defined by (a,b)*(c,d)=(a+c,b+d)
Show that * is commutative and associative. Find theidentity element for * on A, if any.
Answer - 11 : -
A= N x N Binary operation * is defined as (a, b) * (c, d) = (a + c, b + d)
(a) Now (c, d) * (a,b) = (c+a, d+b) = (a+c,b+d)
=> (a, b) * (c, d) = (c, d) * (a, b)
тИ┤ This operation * is commutative
(b) Next(a,b)* [(c,d)*(e,f)]=(a,b)*(c+e,d+f) = ((a + c + e), (b + d + f))
and [(a, b) * (c, d)] * (e, f)=(a+c, b+d) * (e, f) = ((a + c + e, b+d + f))
=> (a, b) * [(c, d) * (e, f)] = [(a, b) * (c, d)] * (e,f)
тИ┤ The binary operation given is associative
(c) Identity element does not exists.
Question - 12 : - Statewhether the following statements are true or false. Justify.
(i) For an arbitrary binary operation * on a set N,
a*a=aтИАaтИИN.
(ii) If * is a commutative binary operation on N,then
a * (b * c) = (c * b) * a
Answer - 12 : -
(i)A binary operation on N is defined as
a*a=aтИАaтИИN.
Here operation * is not defined.
тИ┤ Given statement is false.
(ii) * is a binary commutative operation on N. c
* b = b * c
тИ╡ * is commutative
тИ╡ (c * b) * a = (b * c) * a = a * (b * c)
тИ┤ Thus a * (b * c) = (c * b) * a
This statement is true.
Question - 13 : - Considera binary operation * on N defined as a * b = a┬│ + b┬│. Choose the correctanswer.
Answer - 13 : - (a) Is * both associative and commutative?
(b) Is * commutative but not associative?
(c) Is * associative but not commutative?
(d) Is * neither commutative nor associative?
Solution
On┬аN, the operation *is defined as┬аa┬а*┬аb┬а=┬аa3┬а+┬аb3.
For,┬аa,┬аb,тИИ┬аN, we have:
a┬а*┬аb┬а=┬аa3┬а+┬аb3┬а=┬аb3┬а+┬аa3┬а=┬аb┬а*┬аa┬а[Addition is commutative in┬аN]
Therefore,the operation * is commutative.
Itcan be observed that:
тИ┤(1 * 2) * 3 тЙа 1 * (2 * 3) ; where 1, 2, 3 тИИ┬аN
Therefore,the operation * is not associative.
Hence,the operation * is commutative, but not associative. Thus, the correct answeris B.