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# tutorial4 - ERG2020A Tutorial 4 Karnaugh Map(K-map(2 GAL...

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ERG2020A Tutorial 4 Karnaugh Map (K-map) (2) GAL, OPAL & Lab 2

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Answers to question in tutorial 3 Simplify WXYZ + W X Y Z ’ + WY ’ + Y Z ’ + WYZ ’ + W X WXYZ + W X Y Z ’ + WY ’ + Y Z ’ + WYZ ’ + W X = WXYZ + W X + W X Y Z ’ + Y Z ’ + WYZ ’ + WY = XYZ + W X + Y Z ’ + WZ ’ + WY = XYZ + XY + W X + WY ’ + Y Z ’ + WZ = XZ + XY ’ + W X + WY ’ + Y Z ’ + WZ ’ + WX = XZ + XY ’ + X + WY ’ + Y Z ’ + WZ = X + WY ’ + Y Z ’ + WZ “short” eats “long” re-arrange consensus thm. consensus thm. again “short” eats “long”
Minterm & maxterm index x 1 x 2 x 3 Minterm ( m i ) Maxterm ( M i ) 0 0 0 0 x 1 ' x 2 ' x 3 ' x 1 + x 2 + x 3 1 0 0 1 x 1 ' x 2 ' x 3 x 1 + x 2 + x 3 ' 2 0 1 0 x 1 ' x 2 x 3 ' x 1 + x 2 ' + x 3 3 0 1 1 x 1 ' x 2 x 3 x 1 + x 2 ' + x 3 ' 4 1 0 0 x 1 x 2 ' x 3 ' x 1 ' + x 2 + x 3 5 1 0 1 x 1 x 2 ' x 3 x 1 ' + x 2 + x 3 ' 6 1 1 0 x 1 x 2 x 3 ' x 1 ' + x 2 ' + x 3 7 1 1 1 x 1 x 2 x 3 x 1 ' + x 2 ' + x 3 ' Minterm cares about 1’s of functions Maxterm cares about 0’s of functions By definition, m i = M i ’ & m i ’ = M i

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Minterms and Functions m 3 = m (011) = x 1 x 2 x 3 ’ = ( x 1 + x 2 + x 3 )’ = M 3 x 1 ’ + x 2 + x 3 ’ = M (101) = M 5 F = m 0 + m 1 + m 5 = ( m 2 + m 3 + m 4 + m 6 + m 7 )’ = M 2 M 3 M 4 M 6 M 7
Implicant ( m ) / Implicate ( M ) Implicant / implicate is any rectangles that cover 2 n minterms / maxterms e.g. For 2 e.g. For 2 2 minterms, it can be in the form of 1 x 4 or 2 x 2 minterms, it can be in the form of 1 x 4 or 2 x 2 1 1 1 1 1 1 00 01 00 01 11 10 x1x2 x3x4 0 1 3 2 4 5 7 6 1 2 1 3 1 5 1 4 8 9 1 1 1 0 11 10 00 0 0 0 0 0 0 0 0 00 01 01 11 10 x1x2 x3x4 0 1 3 2 4 5 7 6 1 2 1 3 1 5 1 4 8 9 1 1 1 0 11 10

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Prime Implicant (PIs) They are the largest implicant rectangle you can drawn on K-map A prime implicant is prime when there is no other implicant covers it In this example both of PIs are essential too 1 1 1 1 1 1 00 01 00 01 11 10 x1x2 x3x4 0 1 3 2 4 5 7 6 1 2 1 3 1 5 1 4 8 9 1 1 1 0 11 10
Essential Prime Implicants (EPIs) Essential minterm (Emt) is the minterm covered by only 1 PI. The corresponding PI is called essential PI (EPI) Only the red circle includes m 14 the r ed circle is an EPI Only blue circle covers m 1 , m 3 and m 9 the b lue circle is an EPI Green circle is not EPI, because: m 11 , covered by both green circle and blue circle, is not essential m 15 , covered by both green circle and red circle, is not essential 1 1 1 1 1 1 00 01 00 01 11 10 x1x2 x3x4 0 1 3 2 4 5 7 6 1 2 1 3 1 5 1 4 8 9 1 1 1 0 11 10

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