SOLUTION: By giving a proof or a counterexample, determine the truth value of the following
statement.
For every pair of natural numbers x and y such that x > 2y there is a natural number
Algebra ->
Proofs
-> SOLUTION: By giving a proof or a counterexample, determine the truth value of the following
statement.
For every pair of natural numbers x and y such that x > 2y there is a natural number
Log On
Question 1019873: By giving a proof or a counterexample, determine the truth value of the following
statement.
For every pair of natural numbers x and y such that x > 2y there is a natural number z such that
x > z > y.
You can put this solution on YOUR website! Suppose x and y are a pair of natural numbers such that x > 2y
Since y is a natural number, 2y is a natural number
x must be ( 2y + 1 ) or larger to satisfy x > 2y since 2y + 1
is the next larger natural number above 2y.
Let x' = 2y + 1
Let z = 2y
We know y < 2y and 2y < 2y + 1
so y < 2y < 2y + 1
y < z < x'
We know x' < x
so y < z < x or x > z > y
