Assume you asked your favorite AI learning tool $$Is a dollar contributed to a Roth $401$k equivalent to a dollar contributed to a traditional $401$k$?$ and the AI tool responded as follows:

No$,$ a dollar contributed to a Roth $401($k$)$ is not equal to a dollar contributed to a traditional $401($k$)$ after accounting for taxes. The main difference between these two types of retirement accounts is how they are taxed, which affects the after$-$tax value of your contributions. Here's a breakdown of the differences:

Roth $401($k$)$:

Contributions: When you contribute to a Roth $401($k$),$ you use after$-$tax dollars. This means that you've already paid income tax on the money you contribute.

Tax Treatment of Earnings: The earnings and growth in your Roth $401($k$)$ account are tax$-$free. This means that when you withdraw money in retirement, both your contributions and the earnings they have generated are generally tax$-$free, assuming you meet certain requirements $($e$.$g$.,$ you've had the account for at least five years and are age $59(1)/(2)$ or older$).$

Traditional $401($k$)$:

Contributions: Contributions to a traditional $401($k$)$ are made with pre$-$tax dollars. This means that the money you contribute is deducted from your taxable income for the year in which you make the contribution. As a result, you get an immediate tax benefit because your taxable income is reduced.

Tax Treatment of Earnings: In a traditional $401($k$),$ your contributions and any investment gains grow tax$-$deferred. You don't pay taxes on them until you withdraw the money in retirement. When you make withdrawals in retirement, they are subject to ordinary income tax at your then$-$applicable tax rates.

In summary, when you contribute a dollar to a Roth $401($k$),$ you've already paid income tax on that dollar, and the future withdrawals are generally tax$-$free. When you contribute a dollar to a traditional $401($k$),$ you get a tax deduction upfront, but you'll pay income tax on both your contributions and the earnings when you withdraw them in retirement.

Is the AI response correct? Explain.