Question

2. Exercise 18, 19 Page 188. Determine if the following statements are true. If a statement is true, give a proof from the definitions. If a statement is false, give a counterexample.
(a) If r and s are rational numbers, then (r+s)/2 is rational.
(b) For all real numbers a and b, if a < b then a < (a+b)/2 < b

Answer 1

Answer:

a. true b. true

Step-by-step explanation:

(a) If r and s are rational numbers, then (r+s)/2 is rational.

true

rational numbers can be expresed as fractions

let be r=a/b and s=c/d being a,b,c,d  integer numbers

$\frac{r+s}{2} =\frac{\frac{a}{b}+\frac{c}{d} }{2} =\frac{\frac{da+bc}{bd} }{2} =\frac{da+bc}{2bd}$

d.a=e is an integer number because it's the product of two integers

b.c=f is an integer number because it's the product of two integers

e+f=g  is an integer number because it's the sum of two integers

b.d=h is an integer number because it's the product of two integers

2.h=i is an integer number because it's the product of two integers

g/i=j is an integer number because it's the quotient of two integers

then

$\frac{r+s}{2} =\frac{\frac{a}{b}+\frac{c}{d} }{2} =\frac{\frac{da+bc}{bd} }{2} =\frac{da+bc}{2bd}=\frac{e+f}{2h} =\frac{g}{i} =j$

(b) For all real numbers a and b, if a < b then a < (a+b)/2 < b

true

$a < (a+b)/2 < b$

$2a < (a+b) < 2b$

lets analyze 2a < (a+b)

$2a < (a+b) \\2a-a < (a+b)-a\\a < b$

then 2a < (a+b) is true

lets analyze (a+b) < 2b

$(a+b) < 2b\\(a+b)-b < 2b-b\\a< b$

then (a+b) < 2b is true

Answer 2

Answer:

A , C, and D are true

Step-by-step explanation:

I did the question on edge :]