boolean theorems digital logic design cheet sheet .pdf
- 1. Single Variable theorems Single Variable theorems THEOREMs LOGIC DIAGRAM = ⋅1 X = ⋅X X = ⋅ X X = ⋅0 X 0 1 X 2 AND Laws X 3 0 4
- 2. OR Laws THEOREMs LOGIC DIAGRAM OR Laws = +0 X = +1 X = +X X = +X X X 5 1 6 X 7 1 8 A A A = A A = 9
- 3. Multivariable Theorems The theorems presented below invoke more than one variable. ‰Commutative Laws: Commutative laws allow change in position of AND or OR variables. There are two commutative laws. ‰ This law can be extended to any number of variables. For example, ‰ This law can be extended to any number of variables. For example, ‰Associative Laws : The associative laws allow grouping of variables. There are two associative laws. ‰ This law can be extended to any number of variables. For example, A⋅B = B⋅A 10 Commutative law for AND operation A ⋅ B ⋅ C = B ⋅ C ⋅ A = C ⋅ A ⋅ B = B ⋅ A ⋅ C A+B=B+A 11 Commutative law for OR operation A + B + C = B + C + A = C + A + B = B + A + C A(BC)=(AB)C 12 Associative law for AND operation A +( B + C + D) = (A+ B +C) + D =(A + B) + (C + D) A+(B+C)=(A+B)+C 13 Associative law for OR operation
- 4. ‰Distributive Law: The distributive laws allow factoring or multiplying out of expressions. There are two distributive laws. ‰ This law states that ORing of several variables and ANDing the result with a single variable is equivalent to ANDing that single variable with each of the several variables and then ORing the products A (B+C) = AB +AC 14 ≡ A+BC=(A+B) (A+C) 15 RHS = (A + B) (A + C) = AA + AC + BA + BC = A + AC +AB + BC = A(1 + C + B) + BC = A ⋅ 1 + BC = A + BC = LHS ≡ This can be proved algebraically as shown below.
- 5. ‰ This law states that ORing of a variable (A) with the AND of that variable (A) and another variable (B) is equal to that variable itself (A). ‰ Algebraically, we have ‰ Therefore , ‰ This law states that ORing of a variable with the AND of the complement of that variable with another variable, is equal to the ORing of the two variables. ‰ This law can also be proved algebraically as shown below. A + A⋅B = A 16 A + A ⋅ B = A (1 + B) = A . 1 = A A + A ⋅ Any term = A B A B A A + = + 17 A B A B A A + ≡ B A + B A A +
- 6. MULTIVARIABLE THEOREMS MULTIVARIABLE THEOREMS 9 X+Y=Y+X X+Y=Y+X Commutative laws Commutative laws 10 XY=YX XY=YX X+(Y+Z) = (X+Y)+Z = X+Y+Z X+(Y+Z) = (X+Y)+Z = X+Y+Z X X· ·(YZ) = (XY) (YZ) = (XY)· ·Z =XYZ Z =XYZ 11 Associative laws Associative laws 12 13b 13a X(Y+Z) = XY + XZ X(Y+Z) = XY + XZ (W+X) (Y+Z) = WY+XY+WZ+XZ (W+X) (Y+Z) = WY+XY+WZ+XZ Distributive laws Distributive laws 14 X+XY = X X+XY = X Y X Y X X + = + 15
- 7. Proofing Theorem 15: (X+Y) (X+Z)= X+YZ Z) Y)(X (X + + YZ XY XZ XX + + + ⇓ YZ XY XZ X + + + ⇓ YZ XY X + + ⇓ YZ X + ⇓
- 8. Proofing: Theorem 16 AB A + ) B A(1+ ⇓ A(1) ⇓ A ⇓
- 9. Proofing: Theorem 17 Y X X+ Y X+ ( ) A A B A + + ⇓ ( ) 1 B A + ⇓ B A + ⇓ B A AB A + + ⇓ B A A + Applying Theorem 16 to expand A term Factoring out 2nd & 3rd terms 1 X X theorem Applying = + X 1 X theorem Applying = ⋅
- 10. DeMorgan DeMorgan’ ’s Theorems s Theorems ‰ A great mathematician named DeMorgan contributed two of the most important theorems of Boolean algebra. ‰ DeMorgan’s theorems are extremely useful in simplifying expressions in which a product or sum of variables is inverted. The two theorems are ‰ Theorem (18) says that when the OR sum of two variables is inverted, this is the same as inverting each variable individually and then ANDing these inverted variables. ‰ Theorem (19) says that when the And product of two variables is inverted ,this is the same as inverting each variable individually and then ORing them. Y X Y X ⋅ = + 18 Y X Y X + = ⋅ 19
- 11. Implications of De Morgan Implications of De Morgan’ ’s Theorems s Theorems ‰ Let us examine these theorem (18) and (19) from the standpoint of logic circuits. ‰ ‰ First, consider theorem (18) First, consider theorem (18) ‰ The left-hand side of the equation can be viewed as the output of a NOR gate whose inputs are X and Y. ‰ The right-hand side of the equation, on the other hand, is a result of first inverting both X and Y and then putting them through an AND gate. ‰ What this means is that an AND gate with INVERTERS on each of its inputs is equivalent to a NOR gate. ‰ ‰ Now consider theorem (19) Now consider theorem (19) ‰ The left hand side the equation can be implemented by a NAND gate with input X and Y. ‰ The right side can be implemented by first inverting inputs X and Y and then putting them through an OR gate. ‰ What this means is that an OR gate with INVERTERS on each of its inputs is equivalent to a NAND gate. Y X Y X ⋅ = + Y X + Y X ⋅ Y X Y X + = ⋅ Y X ⋅ Y X +
- 12. Why simplification? ( ) ( ) B A C B A X ⋅ ⋅ ⋅ + = ) C B ( A X ⋅ + = C B X ⋅ = B A X ⋅ =
- 13. Design procedures of combinational logic circuit using algebraic method STEP 1 :Set up the truth table STEP 2 :Write the AND term for each case where the output is a 1 STEP 3 :Write the SOP expression for the output STEP 4 :Simplify the output expression STEP 5 :Implement the circuit for the final expression
- 14. STEP 1 :Set up the truth table A B C BC A C B A C AB ABC X 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1