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<p>Your parents are incorrect. How many doors, and which doors, will be opened with each degree depends on industry and economic conditions at the time; of course, your own interests that appear to prefer the computer engineering (logic gates), software, and math topics can determine which doors you want to go through.</p>
<p>Also, in practice, in the computer industry, employers are often flexible about what degrees they want to see (if they care at all) for experienced job applicants (i.e. not those just graduating college or entering the labor force the first time).</p>