If you know the generators, represent each of the elements of your set as a linear combination of generators. Form a (non-square) matrix where row number $i$ consists of coefficients of element number $i$ in your set. Perform integer Gauss elimination procedure (i.e. you are allowed switching two rows, and subtracting/adding one row from/to another row). Eventually you will get a matrix in the row echelon form. Look how many 1's you have on the diagonal. If the number of 1's is the same as the number of generators, your set generates the whole group. Otherwise the answer is "no".
If you know the generators, represent each of the elements of your set as a linear combination of generators. For Form a (non-square) matrix where row number $i$ consists of coefficients of of elemenent element number $i$. i$in your set. Perform integer Gauss elimination procedure (i.e. you are allowed switching two rows, and subtracting/adding one row from from/to another row). Eventualy Eventually you will get a matrix in the row echelon form. Look how many 1's you have on the diagonal. If the number of 1's is the same as the number of generators, your set generates the whole group. Otherwise the answer is "no". 1 If you know the generators, represent each of the elements of your set as a linear combination of generators. For a (non-square) matrix where row number$i$consists of coefficients of of elemenent number$i\$. Perform integer Gauss elimination procedure (i.e. you are allowed switching two rows, and subtracting/adding one row from another row). Eventualy you will get a matrix in the row echelon form. Look how many 1's you have on the diagonal. If the number of 1's is the same as the number of generators, your set generates the whole group. Otherwise the answer is "no".