Question

Which of the following statements is not true?
(A) Both addition and multiplication are associative for whole numbers.
(B) Zero is the identity for multiplication of whole numbers.
(C) Addition and multiplication both are commutative for whole numbers.
(D) Multiplication is distributive over addition for whole numbers.

Answer 1

(A)
Since, an operation $+$ is said to be associative on whole numbers$\left(W\right)$,
if $a+(b+c)=(a+b)+c,\forall a,b,c\in W$ .....(i)
Take, $a=1,b=2,c=3,$we get

$1+(2+3)=(1+2)+3$

$\Rightarrow 1+5=3+3$

$\Rightarrow 1+5=3+3$

$\therefore 6=6$

Thus, if we take any three elements from whole numbers, it must satisfies equation(i).
Therefore, $+$ is associative operation on $W$ .
Similar way, if we take any three whole numbers, it must satisfies the equation(i), $a\times (b\times c)=(a\times b)\times c,\forall a,b,c\in W$ .
Therefore, $\times$ also associative operation on $W$ .
Hence, the given statement true.
(B)
An element $e$ is called identity element, if $a\times e=e\times a=a,\forall a\in A$
The above identity holds only for $e=1$ as $a\times 1=1\times a=a,\forall a\in A$
Thus, $1$ is called identity element with respect to multiplication.
Therefore, zero cannot be an identity element for the multiplication of whole numbers.
Hence, given statement is false.
(C)
An operation $+$ is called commutative on whole numbers$\left(W\right)$,
$\text{if}a+b=b+a,\forall a,b\in W$
Suppose, $a=1,b=2\text{then}$
consider $a+b=1+2$

$=3$

$=2+1$

$=b+a$

$\therefore a+b=b+a,\forall a,b\in W$

Thus, all elements of whole numbers holds the above identity.
Therefore, $+$ is commutative on whole numbers$\left(W\right)$ .
Similar way, all elements of whole numbers obeys the property of
$a\times b=b\times a,\forall a,b\in W$
Therefore, $\times$ is also a commutative operation on whole numbers.
Hence, the given statement is true.
(D)
Multiplication is distributive over addition for whole numbers$\left(W\right)$,
if $a(b+c)=ab+ac,\forall a,b,c\in W$
Take $a=1,b=2,c=3$
Then $a(b+c)=ab+ac$

$\Rightarrow 1(2+3)=(1\times 2)+(1\times 3)$

$\Rightarrow 1\left(5\right)=2+3$

$\Rightarrow 5=5\left(\text{True}\right)$

Thus,

$a(b+c)=ab+ac,\forall a,b,c\in W$

Therefore, multiplication is distributive over addition for whole numbers.

Hence, the given statement is true.