Final answer:
To simplify logical functions using a K-map, populate the map with values corresponding to minterms and don't-care conditions, then group adjacent values to find Prime Implicants and Essential Prime Implicants.
Step-by-step explanation:
How to Use K-map to Simplify the Following Functions
To use a Karnaugh map (K-map) to simplify logical functions, you need to follow several steps for each function, find all Prime Implicants (PIs) and Essential Prime Implicants (EPIs). Let's examine each given function in detail.
i) f(a,b,c,d) = sum(1,3,5,8,9,12,13,14)
Create a 4-variable K-map and populate it with 1s at the locations corresponding to the minterms 1, 3, 5, 8, 9, 12, 13, and 14. Combine adjacent groups of 1s to find all PIs. Among them, identify the EPIs which are PIs that cover minterm(s) that no other PI covers. This simplified expression will be your result.
ii) f(a,b,c) = sum(0,4,5,6)
For a 3-variable K-map, place 1s for minterms 0, 4, 5, and 6. Group the adjacent 1s to find the PIs and then identify any EPIs for the simplification of the function.
iii) f(a,b,c,d) = sum(3,4) + Don't-Care (5,6,9,11,12,13,14)
In a 4-variable K-map, place 1s for minterms 3 and 4 and Xs for the don't-care conditions at locations 5, 6, 9, 11, 12, 13, and 14. Group the 1s and Xs to get PIs. Since don't-care conditions can be used to make larger groups, they help in further simplifying the function. Identify any EPIs among the PIs to finalize the simplified function.
Note that the position of the 1s and Xs in the K-map and how they can be combined to form groups determines the simplified expression. The goal is to cover all the 1s with the fewest and largest groups of adjacent cells possible.