Find the one’s and two’s complement of each of the following positive binary numbers, assuming
a number system with number of bits 𝑛 = 8.
2. Prove each of the following, using only the six postulates and/or Theorems 1-3 of Boolean
algebra (Each step must be justified by listing the applicable postulate or theorem).
a) (𝑋 + 𝑌)(𝑋 + 𝑌′) = 𝑋
b) 𝑀(𝐴 +𝑀′) = 𝑀𝐴
3. Use the absorption theorems (Theorems 4-5-6), plus any other theorems and postulates, to
simplify the following expressions. (The simplified expressions should have as few terms as
a) 𝐴𝐵𝐶 + 𝐴𝐵𝐶′ + 𝐴𝐵′ + 𝐴𝐵′𝐶′ + 𝐴′𝐵𝐶′
b) (𝑋 + 𝑌 + 𝑍′)(𝑌 + 𝑋 + 𝑍)(𝑌′ + 𝑋)(𝑍′ + 𝑌 + 𝑋′)(𝑋 + 𝑌′ + 𝑍′)
4. Apply DeMorgan’s theorem to convert the following expression to one in which no complement
is applied to more than a single variable.
a) (𝑎𝑏′ + 𝑎𝑐 + 𝑏′𝑐)′
b) ((𝑎′ + 𝑏)(𝑎 + 𝑐′)(𝑏 + 𝑐′ + 𝑑))′
c) ((𝑥′ + 𝑦) + (𝑥 + 𝑧))′
5. Find the simplest switching expressions for the following functions. (Minimize the number of
terms and the number of literals in each term.) The final expression may be in either sum of
products or product of sums form.
a) 𝑓(𝐴, 𝐵, 𝐶, 𝐷) = ∑𝑚(0,1,3,6,7,12,13)
b) 𝑓(𝐴, 𝐵, 𝐶, 𝐷) = ∏𝑀(0,1,3,6,7,12,13)
6. Find the truth tables, minterm lists, and maxterm lists for the following switching functions.
a) 𝑓(𝑎, 𝑏, 𝑐) = 𝑏𝑐 + �̅�𝑏
b) 𝑓(𝑊,𝑋, 𝑌, 𝑍) = (𝑊 + �̅� + �̅�)(𝑋 + �̅� + 𝑍)(�̅� + 𝑌)
c) 𝑓(𝑎, 𝑏, 𝑐) = 𝑏𝑐̅ + �̅��̅�̅̅ ̅̅ ̅̅ ̅̅ ̅̅
7. Find a minimal two-level NAND circuit for each of the following switching functions.
a) 𝑓(𝐴, 𝐵, 𝐶) = ∑𝑚(2,5,6,7)
b) 𝑓(𝐴, 𝐵, 𝐶, 𝐷) = ∏𝑀(0,2,5,7,8,10,13,15)
8. Find a minimal two-level NOR circuit for each of the switching functions in problem 7.
9. For the circuit diagram given below:
a. Derive a switching expression for output Y, in terms of A, B, and C.
b. Find the minterms and maxterms of function Y(A,B,C)