Question

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Boolean Functions, Truth Tables, Logic Minimization, Two-Level Forms Consider a boolean function f (a, b, c, d). Suppose that the function is 1 if

• There is a single 1 among the inputs, or
• There is a single 0 among the inputs, or
• There are exactly two 1’s among the inputs

and it is 0 otherwise.

(a) Write down a truth table for the function

(b) Using a Karnaugh map, provide a minimal sum-of-products (AND-OR) expres- sion.

(c) Using a Karnaugh map, provide a minimal product-of-sums (OR-AND) expres- sion.

(d) Provide a minimal NAND-NAND expression

(e) Provide a minimal OR-NAND expression (f) Provide a minimal NOR-OR expression

(g) Provide a minimal NOR-NOR expression (h) Provide a minimal AND-NOR expression

(i) Provide a minimal NAND-AND expression
(j) Provide a AND-XOR expression (with no negations)

Answer 1

(a) Truth table

Input				Output
a	b	c	d	f
0	0	0	0	0
0	0	0	1	1
0	0	1	0	1
0	0	1	1	1
0	1	0	0	1
0	1	0	1	1
0	1	1	0	1
0	1	1	1	1
1	0	0	0	1
1	0	0	1	1
1	0	1	0	1
1	0	1	1	1
1	1	0	0	1
1	1	0	1	1
1	1	1	0	1
1	1	1	1	0

(b) minimal sum-of-products

f=ab' + cd' + bc' + da'

(c) minimal product-of-sums

f ' = abcd + a'b'c'd'

( f ')' = ( abcd + a'b'c'd' )'

f =( a' + b' + c' + d') ( a + b + c + d)

(d) minimal NAND-NAND expression

f = ab' + cd' + bc' + da'

f = ( (ab')' (cd')' (bc')' (da')' )'

(e) minimal OR-NAND expression

f = ab' + cd' + bc' + da'

f = ( (ab')' (cd')' (bc')' (da')' )'

f = ( (a' + b) (c' + d) (b' + c) (d' + a) )'

(f) NOR-OR expression

f = ab' + cd' + bc' + da'

f = ( (ab')' (cd')' (bc')' (da')' )'

f = ( (a' + b) (c' + d) (b' + c) (d' + a) )'

f = (a' + b)' + (c' + d)' + (b' + c)' + (d' + a)'

(g) minimal NOR-NOR expression

f = ( a' + b' + c' + d') ( a + b + c + d)

f =[ {( a' + b' + c' + d' )}' + {( a + b + c + d )}' ]'

(h) AND-NOR expression

f ' = abcd + a'b'c'd'

( f ')' = ( abcd + a'b'c'd' )'

f = ( abcd + a'b'c'd' )'

(i) NAND-AND expression

f ' = abcd + a'b'c'd'

( f ')' = ( abcd + a'b'c'd' )'

f = ( abcd + a'b'c'd' )'

f = ( abcd )' ( a'b'c'd')'