It wouldn’t be advisable for a person who wants to become an engineer to first major in mathematics with the view to deviating at a later date toward engineering. It is true that a lot of mathematic principles apply in the world of engineering, but engineering is far more practical and ‘hands-on’ and one could not major in mathematics and expect to secure employment right away as an engineer.
If, however, you have majored in mathematics and find yourself looking thereafter to pursue a career in engineering, you are certainly ahead of someone who has majored in something completely unrelated. There are some specialized areas of engineering in which a mathematician could work, but you will still have to undergo further training and obtain the relevant qualifications, or at the very least assume an internship or relevant work experience.
Software engineering is a branch of computer science which includes the development and building of computer systems software and applications software.