The colour change in the apple happen because an enzyme in the apple react with the oxygen which is called oxidation. So due to the oxidation the colour of the apple changes to brown when they are chopped

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Apples contain an enzyme called tyrosinase (or polyphenol oxidase). This enzyme reacts with the oxygen in the atmosphere. Oxidation occurs and rust like brown substance forms. 

Also Apples contain phenols with Iron in them. These compounds also get oxidized on reaction with tyrosinase. There are two copper atoms in tyrosinase enzyme. They react with dioxygen and form intermediate compounds that act on the substrate of the enzyme.   The polyphenols change in to melanin pigment with the brown color.

It is also a similar reason for bananas, potatoes also to turn brown when their cut surfaces are exposed to air (oxygen).