Question

Prove that m² = n²If and only if m = n or m = - n

Short answer

m² = n²If and only if m = n or m = - n

Step-by-step solution

Prove  m   =  n or m  =  - n if given m2 = n2 

The purpose is to prove that\({m^2} = {n^2}\)if and only if\(m = n\)or\(m = - n\).

Assume that\({m^2} = {n^2}\)

Claim:\(m = n\)or\(m = - n\).

For that conversely assume that\(m \ne n\)or\(m \ne - n\)

Therefore\(m - n \ne 0\)or\(m + n \ne 0\)

Consider

\(\begin{array}{l}\left( {m - n} \right)\left( {m + n} \right) \ne 0\\{m^2} + mn - nm - {n^2} \ne 0\\{m^2} - {n^2} \ne 0\\{m^2} \ne {n^2}\end{array}\)

This is the contradiction to the assumption \({m^2} = {n^2}\)

Prove that  \({m^2} = {n^2}\)when given that \(m \ne n\)or \(m \ne  - n\)

Conversely suppose that\(m \ne n\)or\(m \ne - n\)

Claim:\({m^2} = {n^2}\)

For that conversely assume that\({m^2} \ne {n^2}\)

Consider,

\(\begin{array}{l}{m^2} - {n^2} \ne 0\\\left( {m + n} \right)\left( {m - n} \right) \ne 0\end{array}\)

So,

\(m + n \ne 0\)And\(m - n \ne 0\)

\(m \ne n\)And\(m \ne - n\)

Which is contradiction to supposition\(m \ne n\)or\(m \ne - n\)

Hence, if\(m = n\)or\(m = - n\)then\({m^2} = {n^2}\).

Thus, \({m^2} = {n^2}\)if and only if \(m = n\)or \(m = - n\)