All Prime Implicants:
PI 1: m(1,5) using don't care m5
Binary: m1=0001, m5=0101
Position: column yz=01, rows wx=00,01 (adjacent)
Common bits: w=0, y=0, z=1
Term: w'y'z
Covers: m1, m5
PI 2: m(1,3) pair
Binary: m1=0001, m3=0011
Position: row wx=00, columns yz=01,11 (adjacent)
Common bits: w=0, x=0, z=1
Term: w'x'z
Covers: m1, m3
PI 3: m(2,3,6,14) using don't care m2 - 2x2 group!
Binary: m2=0010, m3=0011, m6=0110, m14=1110
Position: columns yz=10,11, multiple rows
Common bits: y=1, z varies, but what's common?
Let me check: all have y=1, and looking at positions...
Wait, m2(00,10), m3(00,11), m6(01,10), m14(11,10)
That's not a perfect rectangle. Let me reconsider.
Actually: m2,m3 in row wx=00, m6 in row wx=01, m14 in row wx=11
These don't form a valid 2x2 group (not all adjacent).
PI 4: m(2,6) using don't care m2
Binary: m2=0010, m6=0110
Position: column yz=10, rows wx=00,01 (adjacent)
Common bits: w=0, y=1, z=0
Term: w'yz'
Covers: m2, m6
PI 5: m(6,14) pair
Binary: m6=0110, m14=1110
Position: column yz=10, rows wx=01,11 (adjacent)
Common bits: x=1, y=1, z=0
Term: xyz'
Covers: m6, m14
PI 6: m(3,11) - checking if they can pair (column yz=11)
m3 at (wx=00,yz=11), m11 at (wx=10,yz=11)
In gray code wx: 00,01,11,10 - so 00 and 10 are NOT adjacent
Cannot form this group!
PI 7: m(8,11) - checking if they can pair (row wx=10)
m8 at (wx=10,yz=00), m11 at (wx=10,yz=11)
In gray code yz: 00,01,11,10 - so 00 and 11 are NOT adjacent
Cannot form this group!
PI 8: m(13,15) using don't cares - pair
Binary: m13=1101, m15=1111
Position: row wx=11, columns yz=01,11 (adjacent)
Common bits: w=1, x=1, z=1
Term: wxz
Covers: m13, m15
PI 9: m(8) - singleton
Binary: m8=1000
Cannot pair with any adjacent cells
Term: wx'y'z'
Covers: m8
PI 10: m(11) - singleton
Binary: m11=1011
Cannot pair with any adjacent cells (checked above)
Term: wxy'z
Covers: m11
Finding Essential Prime Implicants:
Essential PIs (cover minterms that no other PI covers):
• m8 only covered by: wx'y'z' → ESSENTIAL
• m11 only covered by: wxy'z → ESSENTIAL
After essentials, need to cover: m1, m3, m6, m14
Options for m1: w'y'z OR w'x'z
Options for m3: w'x'z
Options for m6: w'yz' OR xyz'
Options for m14: xyz'
Solution 1: Use w'x'z (covers 1,3), xyz' (covers 6,14)
→ f1 = w'x'z + xyz' + wx'y'z' + wxy'z
Solution 2: Use w'y'z (covers 1), w'yz' (covers 6), need more for 3 and 14
→ This doesn't lead to minimum (would need 5 terms)
Solution 3: Use w'x'z (covers 1,3), w'yz' (covers 6), xyz' (covers 14)
→ f3 = w'x'z + w'yz' + xyz' + wx'y'z' + wxy'z (5 terms - not minimum)
Wait, let me reconsider combinations...