Question

Simplify the following logical function using Karnaugh Maps. You will write the result as a sum of products. Do not leave blank spaces in the expression. Write the literals of the terms in alphabetical order. For example, instead of writing the term acb 'write ab'c. Write the function starting with the term that has the fewest literals, and then proceeding in ascending order of literals per term. That is, if for example the simplified function has a term with 4 literals, another with 2 literals and two with 3 literals, start with the 2 literals, then the 3 literals, finally the 4 literal ones. For example, if the result is y = acbd + b'a + ab'c + abc 'your answer will be y = ab' + abc '+ ab'c + abcd. If you have two terms with the same number of literals as in this example, write them in alphabetical order. Assume that a literal not negated goes before a negated one in alphabetical order. In this case we have ab'c and abc '. The second term goes first because the b is not negated. Do not include blank spaces in the expression.

y = abd + a’b’c ’+ ad + a’bcd + abc + acd’

Answer 1

The expression consists of 4 literals. To solve it using K-maps we need to convert the expression in the Canonical SOP(with each term having 4 literals) also called Min-Terms. So we expand the expression using Boolean laws and convert it to Canonical form-

y = abd + a’b’c ’+ ad + a’bcd + abc + acd’

-> y= abcd + abc'd + a'b'c'd + a'b'c'd' + abcd + abc'd + ab'cd + ab'c'd + a'bcd + abcd + abcd' + abcd' + ab'cd'

Now we remove the duplicate literals-

-> y = abcd + abc'd + a'b'c'd + a'b'c'd' + ab'cd + ab'c'd + a'bcd + abcd' + ab'cd'

We can solve this using K-maps of size 4X4.

The groups which will be formed as per the K-map numbering format given below are -

Quad 1 - (9,11,13,15) Expression - AD

Quad 2 - (10,11,14,15) Expression - AC

Pair 1 - (0,1) Expression - A'B'C'

Pair 2 - (7,15) Expression - BCD

The reduced expression will be -

y = AD + AC + A'B'C' + BCD