How can I show this language is Regular? (Foundations of computing) Σ(alphabet) contains 3x1 column matrices 
[[0],[0],[0]], [[0],[0],[1]] , [[0],[1],[0]]... [[1],[1],[1]] such that each 
element in the language is a THREE SET
of the columns where the bottom row is the sum of the two top rows

example
[[0],[0],[1]], [[1],[0],[0]] , [[1],[1],[0]] is an element of the Language 

BUT 

[[0],[0],[1]], [[1],[0],[1]] , [[0],[0],[0]] is not an element of the language.
Your question was already asked before, but got no answer: How to prove for string of triplets that it is a Regular Language?. I merely summarize the comments given there.

You need to implement binary addition. That is easier when you consider first the language in reverse, the lower bits first.

Standard letters are (in a more intuitive notation) [00|0], [01|1], [10|1], [11|0]. In the last case however, we realize that we should take a carry to the next position. This can be solved by adding a special state for that case, and adapting the letters accordingly.