The boolean function for a combinational circuit with four inputs is…

Step 1: List the 1-cells from the K-map (P Q R S):

• 0000 (m0), 0010 (m2), 0011 (m3), 0110 (m6), 0111 (m7), 1000 (m8), 1001 (m9), 1010 (m10), 1101 (m13)

Step 2: Find large groups (prime implicants).

• Q' S' : group of four cells where Q=0 and S=0. This covers minterms 0000, 0010, 1000, 1010 (m0, m2, m8, m10).

• P' R : group of four cells where P=0 and R=1. This covers minterms 0010, 0011, 0110, 0111 (m2, m3, m6, m7).

• Two-cell groups for the remaining 1s (for example the pair at P=1,Q=0,R=0 covers m8 and m9, and the pair at P=1,R=0,S=1 covers m9 and m13). These cover the remaining 1-cells not in the above 4-cell groups.

Step 3: Identify essential prime implicants.

• Consider the minterm 0000 (m0). It is covered by Q' S' (Q=0,S=0). There is no other prime implicant that can cover 0000 without also including a 0 in the map, so this minterm is covered only by Q' S'.

• Because at least one minterm (0000) is covered uniquely by Q' S', Q' S' is an essential prime implicant.

Final conclusion: Q' S' is an essential prime implicant.