J.R. S. answered 03/13/18
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Ph.D. in Biochemistry--University Professor--Chemistry Tutor
An redox (oxidation/reduction) reaction is one in which one element loses electrons and another gains electrons.
A synthesis reaction is of the form A + B ==> C.
Take the example of sodium plus chlorine combining to make sodium chloride.
2Na(s) + Cl2(g) ==> 2NaCl(s)
This is clearly a synthesis reaction (A + B ==>C). But note that each Na(s) loses an electron to become Na+ and each Cl gains an electron to become Cl-, so this is clearly an oxidation/reduction reaction.
So, to answer the question, a redox reaction can also be a synthesis reaction (if there is a giving and taking of electrons), and redox is also single displacement and combustion. But redox is never double displacement reactions.
Hope this makes sense.