This Discrete Math Please show all the steps Prove that the

This Discrete Math:

Please show all the steps.

Prove that the two definitions of the rational numbers below are equivalent. Q_1 = {n/m: n, m element Z and m notequalto 0} and Q_2 = {a/b: a element Z, b element N and gcd(a, b) = 1}

Solution

a number is rational if we can write it in a form p/q where p and q are integers, and q is not zero. Clearly all fractions are of that form, so fractions are rational numbers.

=> Rational numbers are whole numbers, fractions, and decimals

In the first definition n and m are integres and m is not equation to 0 .

and even if n is divisible by m completly and we get an integer quotient and the remainder is 0 . Even then it will be termed as a rational number as if n/m = c , where c is an integer

then c = c/1 and c/1 is a rational number

in the second a and b are integers and there is nothing common between a and b besides 1 .

and we already know that 0 is not an integer

so b will never be = 0

Hence Q1 and Q2 are equivalent.