Question

Proofs

For this assignment, know that:

An integer is any countable number. Examples are: -3, 0, 5, 1337, etc.

A rational number is any number that can be written in the form a/b, a and b are integers in lowest terms, and b cannot equal 0. Examples are 27, 22/7, -3921/2, etc.

A real number is any number that is not imaginary or infinity. Examples are 0, 4/3, square root of 2, pi, etc.

1. Prove by cases that for all real numbers | x + y | <= |x| + |y|
2. Prove by contradiction that the average of three real numbers is greater than or equal to at least one of the numbers

Answer 1

Prove by cases that for all real numbers |x+y|<=|x| + |y|

if all know that |x| will always give us a positive value no matter what or we can say |x| = +x
|x| = for all values x is greater than 0 where x !=0

so let's take some examples

case 1: x = -3 and y = 2
|x+y| = |-3+2| = |-1| = 1
|x|+|y| = |-3| +| |2| = 3+2 = 5

case 2 :when both positive than we know value would be same
i.e |x+y| = |x| + |y|

case 3: when both values are negative then also value would be the same for both
   for eg : x = -3 and y = -2
|x+y| = |-3+-2| = |-5| = 5
|x| + |y| = |-3|+ |-2| = 3+2 = 5

so this proves that |x+y|<=|x| + |y|

Prove by contradiction that the average of three real numbers is greater than or equal to at least one of the numbers

let's understand first what average do :
average of any numbers will give the answer between or equal to the min value and max value from the list of values

for example x = 1, y=2 ,z = 3
so avg = (1+2+3)/3
avg = 2

so we see the value exist between min value i.e 1 and max value ie 3

so let's understand it by contradiction

it starts with negation and says that:-
suppose we have three real numbers x,y,z such that their average is
maximum than all three numbers

so let m is the average of three number

then
(x+y+z)/3 = m ----(line 1)
according to our assumption (x<m) ,(y<m) and (z<m)

adding all inequalities we get x+y+x<3m.

we know that m is defined as in-line (1) as x+y+z = 3m

but now 3m = x+y+z < 3m. So 3m<3m which is not possible so our assumption is wrong so this proves that the average of three real numbers is greater than or equal to at least one of the numbers

lets take one more example : (enjoying right .....(* ~*)
lets say x = 5 ,y = 6,and z = 8

avg = (4+6+8)/3
avg = 6
so avg is less than or equal to 6 and 8

if any doubt comment below