Total batters = 19
Let right-handed batters = r
Lef left-handed batters = l
Let both left and right-handed batters = b
Question=> How many of the 19 batters bat left-handed?
Total left-handed batters = l + b
Statement 1: 7 of the batters bat right-handed but do not bat left-handed
Total batters = r+l+b
19 = r+l+b where r=7
19 = 7 (l+b)
l + b = 19 - 7 = 12 where (l+b) = total left-handed batters
Therefore, total left-handed batters = 12; hence, statement 1 is SUFFICIENT.

Statement 2: 4 of the batters bat both right-handed and left-handed
This means that b=4
Total batters = r+l+b
19 = r+l+4
r+l = 19-4 = 15
This statement cannot be used to evaluate l+b because the value of l is unknown, hence; statement 2 is NOT SUFFICIENT.

Since only statement 1 is SUFFICIENT, answer = option A