Chapter 1.4, Problem 14GP

To determine

Whether the given conjecture is true. If not, then give a counterexample.

Answer to Problem 14GP

  $\text{Not true}$

Explanation of Solution

Given information:

The sum of two odd numbers is always odd.

Concept Used:

Commutative property:

  $a+b=b+a$

  $a\cdot b=b\cdot a$

Associative property:

  $\left(a+b\right)+c=a+\left(b+c\right)$

  $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$

Distributive property:

  $a\left(b+c\right)=ab+ac$

  $\left(b+c\right)a=ba+ca$

Additive identity:

  $a+0=0+a=a$

Multiplicative identity:

  $a\cdot 1=1\cdot a=a$

Zero property of Multiplication:

  $a\cdot 0=0\cdot a=0$

Calculation:

The sum of two odd numbers is always even. For example,

  $\begin{array}{c}1+3=4,\\ 5+9=14,\\ 11+7=18,\\ 7+13=20\text{etc}\mathrm{..}\end{array}$

While adding two odd numbers, the sum will be equal to an even number.

Thus, the conjecture that the sum of two odd numbers is always odd is not true.