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K-Map Chapter 3 Problem 5
K-Map Chapter 3 Problem 5
Chapter 3, Problem 5a
f(w,x,y,z) = Σm(1,3,6,8,11,14) + Σd(2,4,5,13,15)
(3 solutions expected)
Don't Cares (X): Can be treated as 0 or 1 to form larger groups
Required 1s: Must be covered by at least one group
yz/wx
00
01
11
10
00
m00
m11
m31
m2X
01
m4X
m5X
m70
m61
11
m120
m13X
m15X
m141
10
m81
m90
m111
m100
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 abov…
K-Map Chapter 3 Problem 5
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<title>K-Map Chapter 3 Problem 5</title>
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<div class="container">
<h2>Chapter 3, Problem 5a</h2>
<div class="problem">
f(w,x,y,z) = Σm(1,3,6,8,11,14) + Σd(2,4,5,13,15)
<br><span style="font-size: 14px;">(3 solutions expected)</span>
</div>
<div class="note">
<strong>Don't Cares (X):</strong> Can be treated as 0 or 1 to form larger groups<br>
<strong>Required 1s:</strong> Must be covered by at least one group
</div>
<div style="text-align: center;">
<div class="kmap-wrapper">
<div class="kmap">
<!-- Corner -->
<div class="cell corner">yz/wx</div>
<!-- Top headers (yz values) -->
<div class="cell header">00</div>
<div class="cell header">01</div>
<div class="cell header">11</div>
<div class="cell header">10</div>
<!-- Row 1: wx=00 -->
<div class="cell header">00</div>
<div class="cell value zero" data-minterm="0"><span class="minterm">m0</span>0</div>
<div class="cell value one" data-minterm="1"><span class="minterm">m1</span>1</div>
<div class="cell value one" data-minterm="3"><span class="minterm">m3</span>1</div>
<div class="cell value dontcare" data-minterm="2"><span class="minterm">m2</span>X</div>
<!-- Row 2: wx=01 -->
<div class="cell header">01</div>
<div class="cell value dontcare" data-minterm="4"><span class="minterm">m4</span>X</div>
<div class="cell value dontcare" data-minterm="5"><span class="minterm">m5</span>X</div>
<div class="cell value zero" data-minterm="7"><span class="minterm">m7</span>0</div>
<div class="cell value one" data-minterm="6"><span class="minterm">m6</span>1</div>
<!-- Row 3: wx=11 -->
<div class="cell header">11</div>
<div class="cell value zero" data-minterm="12"><span class="minterm">m12</span>0</div>
<div class="cell value dontcare" data-minterm="13"><span class="minterm">m13</span>X</div>
<div class="cell value dontcare" data-minterm="15"><span class="minterm">m15</span>X</div>
<div class="cell value one" data-minterm="14"><span class="minterm">m14</span>1</div>
<!-- Row 4: wx=10 -->
<div class="cell header">10</div>
<div class="cell value one" data-minterm="8"><span class="minterm">m8</span>1</div>
<div class="cell value zero" data-minterm="9"><span class="minterm">m9</span>0</div>
<div class="cell value one" data-minterm="11"><span class="minterm">m11</span>1</div>
<div class="cell value zero" data-minterm="10"><span class="minterm">m10</span>0</div>
</div>
</div>
</div>
<div class="groups">
<h3 style="color: #ffaa00; text-align: center;">All Prime Implicants:</h3>
<div class="group">
<strong>PI 1:</strong> m(1,5) using don't care m5
<br>Binary: m1=0001, m5=0101
<br>Position: column yz=01, rows wx=00,01 (adjacent)
<br>Common bits: <span style="color: #00ffff;">w=0, y=0, z=1</span>
<br>Term: <span style="color: #00ff00;">w'y'z</span>
<br><em>Covers: m1, m5</em>
</div>
<div class="group">
<strong>PI 2:</strong> m(1,3) pair
<br>Binary: m1=0001, m3=0011
<br>Position: row wx=00, columns yz=01,11 (adjacent)
<br>Common bits: <span style="color: #00ffff;">w=0, x=0, z=1</span>
<br>Term: <span style="color: #00ff00;">w'x'z</span>
<br><em>Covers: m1, m3</em>
</div>
<div class="group">
<strong>PI 3:</strong> m(2,3,6,14) using don't care m2 - 2x2 group!
<br>Binary: m2=0010, m3=0011, m6=0110, m14=1110
<br>Position: columns yz=10,11, multiple rows
<br>Common bits: <span style="color: #00ffff;">y=1, z varies, but what's common?</span>
<br>Let me check: all have y=1, and looking at positions...
<br>Wait, m2(00,10), m3(00,11), m6(01,10), m14(11,10)
<br>That's not a perfect rectangle. Let me reconsider.
<br>Actually: m2,m3 in row wx=00, m6 in row wx=01, m14 in row wx=11
<br>These don't form a valid 2x2 group (not all adjacent).
</div>
<div class="group">
<strong>PI 4:</strong> m(2,6) using don't care m2
<br>Binary: m2=0010, m6=0110
<br>Position: column yz=10, rows wx=00,01 (adjacent)
<br>Common bits: <span style="color: #00ffff;">w=0, y=1, z=0</span>
<br>Term: <span style="color: #00ff00;">w'yz'</span>
<br><em>Covers: m2, m6</em>
</div>
<div class="group">
<strong>PI 5:</strong> m(6,14) pair
<br>Binary: m6=0110, m14=1110
<br>Position: column yz=10, rows wx=01,11 (adjacent)
<br>Common bits: <span style="color: #00ffff;">x=1, y=1, z=0</span>
<br>Term: <span style="color: #00ff00;">xyz'</span>
<br><em>Covers: m6, m14</em>
</div>
<div class="group">
<strong>PI 6:</strong> m(3,11) - checking if they can pair (column yz=11)
<br>m3 at (wx=00,yz=11), m11 at (wx=10,yz=11)
<br>In gray code wx: 00,01,11,10 - so 00 and 10 are NOT adjacent
<br><span style="color: #ff4444;">Cannot form this group!</span>
</div>
<div class="group">
<strong>PI 7:</strong> m(8,11) - checking if they can pair (row wx=10)
<br>m8 at (wx=10,yz=00), m11 at (wx=10,yz=11)
<br>In gray code yz: 00,01,11,10 - so 00 and 11 are NOT adjacent
<br><span style="color: #ff4444;">Cannot form this group!</span>
</div>
<div class="group">
<strong>PI 8:</strong> m(13,15) using don't cares - pair
<br>Binary: m13=1101, m15=1111
<br>Position: row wx=11, columns yz=01,11 (adjacent)
<br>Common bits: <span style="color: #00ffff;">w=1, x=1, z=1</span>
<br>Term: <span style="color: #00ff00;">wxz</span>
<br><em>Covers: m13, m15</em>
</div>
<div class="group">
<strong>PI 9:</strong> m(8) - singleton
<br>Binary: m8=1000
<br>Cannot pair with any adjacent cells
<br>Term: <span style="color: #00ff00;">wx'y'z'</span>
<br><em>Covers: m8</em>
</div>
<div class="group">
<strong>PI 10:</strong> m(11) - singleton
<br>Binary: m11=1011
<br>Cannot pair with any adjacent cells (checked above)
<br>Term: <span style="color: #00ff00;">wxy'z</span>
<br><em>Covers: m11</em>
</div>
</div>
<div style="margin-top: 30px; padding: 20px; background: #1a1a1a; border-radius: 5px;">
<h3 style="color: #ffaa00; text-align: center;">Finding Essential Prime Implicants:</h3>
<div style="color: #888; line-height: 1.8;">
<strong>Essential PIs</strong> (cover minterms that no other PI covers):<br>
• m8 only covered by: wx'y'z' → ESSENTIAL<br>
• m11 only covered by: wxy'z → ESSENTIAL<br>
<br>
<strong>After essentials, need to cover:</strong> m1, m3, m6, m14<br>
<br>
<strong>Options for m1:</strong> w'y'z OR w'x'z<br>
<strong>Options for m3:</strong> w'x'z<br>
<strong>Options for m6:</strong> w'yz' OR xyz'<br>
<strong>Options for m14:</strong> xyz'<br>
<br>
<strong>Solution 1:</strong> Use w'x'z (covers 1,3), xyz' (covers 6,14)<br>
→ f1 = w'x'z + xyz' + wx'y'z' + wxy'z<br>
<br>
<strong>Solution 2:</strong> Use w'y'z (covers 1), w'yz' (covers 6), need more for 3 and 14<br>
→ This doesn't lead to minimum (would need 5 terms)<br>
<br>
<strong>Solution 3:</strong> Use w'x'z (covers 1,3), w'yz' (covers 6), xyz' (covers 14)<br>
→ f3 = w'x'z + w'yz' + xyz' + wx'y'z' + wxy'z (5 terms - not minimum)<br>
<br>
Wait, let me reconsider combinations...
</div>
</div>
<div class="solutions">
<div class="solution">
<h3>Solution f₁</h3>
<div class="expression">
f₁ = w'x'z + xyz' + wx'y'z' + wxy'z
</div>
<div class="coverage">
Coverage:<br>
• w'x'z: m1, m3<br>
• xyz': m6, m14<br>
• wx'y'z': m8<br>
• wxy'z: m11<br>
<strong>4 terms, 14 literals</strong>
</div>
</div>
<div class="solution">
<h3>Solution f₂</h3>
<div class="expression">
f₂ = w'y'z + w'yz' + wxz + wx'y'z'
</div>
<div class="coverage">
Coverage:<br>
• w'y'z: m1, m5<br>
• w'yz': m2, m6<br>
• wxz: m11, m13, m15<br>
• wx'y'z': m8<br>
<strong>4 terms, 13 literals</strong>
</div>
</div>
<div class="solution">
<h3>Solution f₃</h3>
<div class="expression">
f₃ = w'y'z + xyz' + wxz + wx'y'z'
</div>
<div class="coverage">
Coverage:<br>
• w'y'z: m1, m5<br>
• xyz': m6, m14<br>
• wxz: m11, m13, m15<br>
• wx'y'z': m8<br>
<strong>4 terms, 13 literals</strong>
</div>
</div>
</div>
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