Sure, let's carefully examine each statement one by one to determine their validity.
### Statement (i): The sum of two odd whole numbers is always a whole number
Consider any two odd whole numbers. An odd number can be represented in the form [tex]\(2k + 1\)[/tex], where [tex]\(k\)[/tex] is an integer.
1. Let the first odd number be [tex]\(2a + 1\)[/tex].
2. Let the second odd number be [tex]\(2b + 1\)[/tex].
3. The sum of these two odd numbers is:
[tex]\[
(2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1)
\][/tex]
The result is [tex]\(2(a + b + 1)\)[/tex], which is clearly a whole number since it is an integer.
Therefore, the sum of two odd whole numbers is always a whole number, making Statement (i) true.
### Statement (ii): The product of two odd numbers is always odd
Let's check this by again taking two odd numbers, which can be represented as [tex]\(2k + 1\)[/tex], where [tex]\(k\)[/tex] is an integer.
1. Let the first odd number be [tex]\(2a + 1\)[/tex].
2. Let the second odd number be [tex]\(2b + 1\)[/tex].
3. The product of these two odd numbers is:
[tex]\[
(2a + 1) \times (2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1
\][/tex]
The result is of the form [tex]\(2m + 1\)[/tex] where [tex]\(m\)[/tex] is an integer, which is again an odd number.
Therefore, the product of two odd numbers is always odd, making Statement (ii) true.
### Statement (iii): The product of two even numbers is always even
Now let’s verify this using two even numbers. An even number can be represented as [tex]\(2k\)[/tex], where [tex]\(k\)[/tex] is an integer.
1. Let the first even number be [tex]\(2a\)[/tex].
2. Let the second even number be [tex]\(2b\)[/tex].
3. The product of these two even numbers is:
[tex]\[
(2a) \times (2b) = 4ab = 2(2ab)
\][/tex]
The result is of the form [tex]\(2n\)[/tex] where [tex]\(n\)[/tex] is an integer, which is clearly an even number.
Therefore, the product of two even numbers is always even, making Statement (iii) true.
### Conclusion
Each of the three statements is true:
(i) The sum of two odd whole numbers is always a whole number.
(ii) The product of two odd numbers is always odd.
(iii) The product of two even numbers is always even.
