Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table. (i) Compute (2 * 3) * 4 and 2 * (3 * 4) (ii) Is * commutative? (iii) Compute (2 * 3) * (4 * 5). (Hint: use the following 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

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Solution

(i)

The table of the operation is shown below:

* 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

The value of the function ( 23 )4 from the table is,

( 23 )4=14 =1

The value of the function 2( 34 ) from the table is,

2( 34 )=21 =1

Thus, the value of ( 23 )4 is 1 and the value of 2( 34 ) is 1 .

(ii)

For the function to be commutative ab=ba .

The function for any random value of a and b be a=4 and b=2 is,

42=2 24=2

So for every value of a,b∈{ 1,2,3,4,5 } , the function is commutative.

Thus, the function is commutative.

(iii)

The value of the function ( 23 )( 45 ) from the table is,

( 23 )( 45 )=11 =1

Thus, the value of ( 23 )( 45 ) is 1 .

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Similar questions

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

(i) Compute (2 * 3) * 4 and 2 * (3 * 4)

(ii) Is * commutative?

(iii) Compute (2 * 3) * (4 * 5).

(Hint: use the following 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

Consider a binary operation $*$ on set{1,2,3,4,5} given by the following multiplication table:
(iii)Compute $(2 * 3) \times (4 * 5)$
$\begin{matrix} * & 1 & 2 & 3 & 4 & 5 1 & 1 & 1 & 1 & 1 & 1 2 & 1 & 2 & 1 & 2 & 1 3 & 1 & 1 & 3 & 1 & 1 4 & 1 & 2 & 1 & 4 & 1 5 & 1 & 1 & 1 & 1 & 5 \end{matrix}$

Consider a binary operation $*$ on set{1,2,3,4,5} given by the following multiplication table:
(ii) Is $*$ commutative?

$\begin{matrix} * & 1 & 2 & 3 & 4 & 5 1 & 1 & 1 & 1 & 1 & 1 2 & 1 & 2 & 1 & 2 & 1 3 & 1 & 1 & 3 & 1 & 1 4 & 1 & 2 & 1 & 4 & 1 5 & 1 & 1 & 1 & 1 & 5 \end{matrix}$

Consider a binary operation $*$ on set{1,2,3,4,5} given by the following multiplication table:
(i)Compute $(2 * 3) * 4$ and $2 * (3 * 4)$

$\begin{matrix} * & 1 & 2 & 3 & 4 & 5 1 & 1 & 1 & 1 & 1 & 1 2 & 1 & 2 & 1 & 2 & 1 3 & 1 & 1 & 3 & 1 & 1 4 & 1 & 2 & 1 & 4 & 1 5 & 1 & 1 & 1 & 1 & 5 \end{matrix}$

(ii) Is $*$ commutative?

$\begin{matrix} * & 1 & 2 & 3 & 4 & 5 1 & 1 & 1 & 1 & 1 & 1 2 & 1 & 2 & 1 & 2 & 1 3 & 1 & 1 & 3 & 1 & 1 4 & 1 & 2 & 1 & 4 & 1 5 & 1 & 1 & 1 & 1 & 5 \end{matrix}$

(iii)Compute $(2 * 3) \times (4 a s t 5)$
$\begin{matrix} * & 1 & 2 & 3 & 4 & 5 1 & 1 & 1 & 1 & 1 & 1 2 & 1 & 2 & 1 & 2 & 1 3 & 1 & 1 & 3 & 1 & 1 4 & 1 & 2 & 1 & 4 & 1 5 & 1 & 1 & 1 & 1 & 5 \end{matrix}$

Q. Consider the binary operation

\land

on the set

{1, 2, 3, 4, 5}

defined by

a \land b = m i n {a, b}

. Write the operation table of the operation

\land

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