Answer:
A, B, D and E
Step-by-step explanation:
Given
Options A to F
Required
Determine which is true
Option A:
This is always true;
Take for instance the following rational numbers
[tex]a = \frac{1}{2}[/tex] [tex]b = \frac{1}{3}[/tex]
[tex]a + b = \frac{1}{2} + \frac{1}{3}[/tex]
[tex]a + b = \frac{3 + 2}{6}[/tex]
[tex]a + b = \frac{5}{6}[/tex]
This will always result in a rational number
Option B:
This is always true;
Take for instance the following rational number
[tex]a = 0.5[/tex]
And the following irrational number
[tex]b = 3.142857[/tex]
[tex]a + b =0.5+ 3.142857[/tex]
[tex]a + b =3.642857[/tex]
This will always result in an irrational number
Option C:
This is not always true;
1. Take for instance the following irrational numbers
[tex]a = 0.33333[/tex] [tex]b = 3.142857[/tex]
[tex]a + b =0.33333 + 3.142857[/tex]
[tex]a + b =3.476187[/tex]
2. Take for instance the following irrational numbers
[tex]a = 3 + \sqrt5[/tex] [tex]b = -\sqrt5[/tex]
[tex]a + b = 3 + \sqrt5 - \sqrt5[/tex]
[tex]a + b = 3[/tex]
From the above examples, this implies that the statement is not always true
D.
This is always true;
Take for instance the following rational numbers
[tex]a = \frac{1}{2}[/tex] [tex]b = \frac{1}{3}[/tex]
[tex]a * b = \frac{1}{2} * \frac{1}{3}[/tex]
[tex]a * b = \frac{1}{6}[/tex]
This will always result in a rational number
E.
This is always true;
Take for instance the following rational number
[tex]a = 0.5[/tex]
And the following irrational number
[tex]b = 3.142857[/tex]
[tex]a * b =0.5 * 3.142857[/tex]
[tex]a * b =1.5714285[/tex]
This will always result in an irrational number
F.
This is not always true;
1. Take for instance the following irrational numbers
[tex]a = 0.33333[/tex] [tex]b = 3.142857[/tex]
[tex]a * b =0.33333 * 3.142857[/tex]
[tex]a * b =1.04760852381[/tex]
2. Take for instance the following irrational numbers
[tex]a = 3 + \sqrt5[/tex] [tex]b = 0[/tex]
[tex]a * b = (3 + \sqrt5) * 0[/tex]
[tex]a * b = 0[/tex]
From the above examples, this implies that the statement is not always true
