boolean logic - sum of minterm vs product of maxterm -

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given following boolean expression of f(a,b,c): f(a,b,c) = a' + b + c' of following statements is/are true above expression?

(i) sop expression (ii) pos expression (iii) sum-of-minterms expression (iv) product-of-maxterms expression

the model answer question i),ii) , iv)

my question why iii) not 1 of answers? drew k-map , found out possible derive such sum-of-minters expression

a cluster of literals in boolean expression forms minterm or maxterm only, if there literals (variables of given function or negation) included in it.

a minterm product of literals of function, maxterm sum of literals of function.

in k-map minterm or maxterm marks out 1 cell. in truth table maxterm or minterm matches 1 row.

the following truth-table corresponds given function:

 index | | b | c || f(a,b,c) | term matching row/k-map cell -------|---|---|---||----------|----------------------------------    0   | 0 | 0 | 0 ||     1    | minterm: m0 = (¬a⋅¬b⋅¬c)    1   | 0 | 0 | 1 ||     1    | minterm: m1 = (¬a⋅¬b⋅c)    2   | 0 | 1 | 0 ||     1    | minterm: m2 = (¬a⋅b⋅¬c)    3   | 0 | 1 | 1 ||     1    | minterm: m3 = (¬a⋅b⋅c) -------|---|---|---||----------|----------------------------------    4   | 1 | 0 | 0 ||     1    | minterm: m4 = (a⋅¬b⋅¬c)    5   | 1 | 0 | 1 ||     0    | maxterm: m5 = (¬a + b + ¬c)    6   | 1 | 1 | 0 ||     1    | minterm: m6 = (a⋅b⋅¬c)    7   | 1 | 1 | 1 ||     1    | minterm: m7 = (a⋅b⋅c)

there 1 maxterm present in truth table (and k-map) , maxterm determining function's output logical 0. valid product-of-maxterms expression, if there one. same boolean expression original one, valid product-of-maxterms expression too.

however, not valid sum of minterms, because there none:

f(a,b,c) = ∏(5) = m5 = (¬a + b + ¬c)

for original expression sum of minterms, need mark out every single true/one cell in k-map separately this:

f(a,b,c) = ∑(0,1,2,3,4,6,7) = m0 + m1 + m2 + m3 + m4 + m6 + m7 =          = (¬a⋅¬b⋅¬c)+(¬a⋅¬b⋅c)+(¬a⋅b⋅¬c)+(¬a⋅b⋅c)+(a⋅¬b⋅¬c)+(a⋅b⋅¬c)+(a⋅b⋅c)

as can see, if these 2 boolean expressions equivalent each other, original 1 (on left side of equation) not written sum-of-minterms expression (on right side of equation).

(¬a+b+¬c) = (¬a⋅¬b⋅¬c)+(¬a⋅¬b⋅c)+(¬a⋅b⋅¬c)+(¬a⋅b⋅c)+(a⋅¬b⋅¬c)+(a⋅b⋅¬c)+(a⋅b⋅c)

just any product not minterm, original expression in form of both product of sum , sum of products, not valid sum-of-minterms.

f(a,b,c) = (¬a + b + ¬c) = (¬a) + (b) + (¬c)

in picture (created using latex) can see expression – same in it's minimal dnf , minimal cnf – , sum of minterms equivalent it.

k-maps equivalent expressions.


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