BOOLEAN
- 1. Fredrick Kendrick Electrical Engineering Dept. ET1220 Boolean Algebra Inputs Truth Table Outputs For Each Gate A B AND NAND OR NOR EX-OR EX-NOR 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 0 1
- 2. Fredrick Kendrick Electrical Engineering Dept. ET1220 The following table gives a list of the common logic functions and their equivalent Boolean notation. Logic Function Boolean Notation AND A.B OR A+B NOT A NAND A .B NOR A+B EX-OR (A.B) + (A.B) or A B EX-NOR (A.B) + or A B 2-input logic gate truth tables are given here as examples of the operation of each logic function, but there are many more logic gates with 3, 4 even 8 individual inputs. The multiple input gates are no different to the simple 2-input gates above, so a 4-input AND gate would still require ALL 4-inputs to be present to produce the required output at Q and its larger truth table would reflect that. Truth Tables for the Laws of Boolean
- 3. Fredrick Kendrick Electrical Engineering Dept. ET1220 Boolean Expression Description Equivalent Switching Circuit Boolean Algebra Law or Rule A + 1 = 1 A in parallel with closed = "CLOSED" Annulment A + 0 = A A in parallel with open = "A" Identity A . 1 = A A in series with closed = "A" Identity A . 0 = 0 A in series with open = "OPEN" Annulment A + A = A A in parallel with A = "A" Indempotent A . A = A A in series with A = "A" Indempotent NOT A = A NOT NOT A (double negative) = "A" Double Negation A + A = 1 A in parallel with not A Complement
- 4. Fredrick Kendrick Electrical Engineering Dept. ET1220 = "CLOSED" A . A = 0 A in series with not A = "OPEN" Complement A+B = B+A A in parallel with B = B in parallel with A Commutative A.B = B.A A in series with B = B in series with A Commutative A+B = A.B invert and replace OR with AND de Morgan’s Theorem A.B = A+B invert and replace AND with OR de Morgan’s Theorem The basic Laws of Boolean Algebra that relate to the Commutative Law allowing a change in position for addition and multiplication, the Associative Law allowing the removal of brackets for addition and multiplication, as well as the distributive Law allowing the factoring of an expression, are the same as in ordinary algebra. Each of the Boolean Laws above are given with just a single or two variables, but the number of variables defined by a single law is not limited to this as there can be an infinite number of variables as inputs too the expression. These Boolean laws detailed above can be used to prove any given Boolean expression as well as for simplifying complicated digital circuits. A brief description of the various Laws of Boolean are given below with A representing a variable input.
- 5. Fredrick Kendrick Electrical Engineering Dept. ET1220 Description of the Laws of Boolean Algebra Annulment Law – A term AND´ed with a “0” equals 0 or OR´ed with a “1” will equal 1. o A . 0 = 0, A variable AND’ed with 0 is always equal to 0. o A + 1 = 1, A variable OR’ed with 1 is always equal to 1. Identity Law – A term OR´ed with a “0” or AND´ed with a “1” will always equal that term. o A + 0 = A, A variable OR’ed with 0 is always equal to the variable. o A . 1 = A, A variable AND’ed with 1 is always equal to the variable. Indempotent Law – An input AND´ed with itself or OR´ed with itself is equal to that input. o A + A = A, A variable OR’ed with itself is always equal to the variable. o A . A = A, A variable AND’ed with itself is always equal to the variable. Complement Law – A term AND´ed with its complement equals “0” and a term OR´ed with its complement equals “1”. o A . A = 0, A variable AND’ed with its complement is always equal to 0. o A + A = 1, A variable OR’ed with its complement is always equal to 1. Commutative Law – The order of application of two separate terms is not important. o A . B = B . A, The order in which two variables are AND’ed makes no difference. o A + B = B + A, The order in which two variables are OR’ed makes no difference. Double Negation Law – A term that is inverted twice is equal to the original term. o A = A, A double complement of a variable is always equal to the variable. de Morgan´s Theorem – There are two “de Morgan´s” rules or theorems, (1) Two separate terms NOR´ed together is the same as the two terms inverted (Complement) and AND´ed for example, A+B = A . B. (2) Two separate terms NAND´ed together is the same as the two terms inverted (Complement) and OR´ed for example, A . B = A + B.
- 6. Fredrick Kendrick Electrical Engineering Dept. ET1220 Other algebraic laws not detailed above include: Distributive Law – This law permits the multiplying or factoring out of an expression. Absorptive Law – This law enables a reduction in a complicated expression to a simpler one by absorbing like terms. Associative Law – This law allows the removal of brackets from an expression and regrouping of the variables. Boolean algebra Functions Using the information above, simple 2-input AND, OR and NOT Gates can be represented by 16 possible functions as shown in the following table. FunctionDescription Expression 1. NULL 0 2. IDENTITY 1 3. Input A A 4. Input B B 5. NOT A A 6. NOT B B 7. A AND B (AND) A . B 8. A AND NOT B A . B 9. NOT A AND B A . B 10. NOT A AND NOT B (NAND)A . B 11. A OR B (OR) A + B 12. A OR NOT B A + B 13. NOT A OR B A + B
- 7. Fredrick Kendrick Electrical Engineering Dept. ET1220 14. NOT OR (NOR) A + B 15. Exclusive-OR A.B + A.B 16. Exclusive-NOR A.B + A.B Laws of Boolean Algebra Example No1 Using the above laws, simplify the following expression: (A + B) (A + C) ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| Truth Tables for the Laws of Boolean Boolean Expression Description Equivalent Switching Circuit Boolean Algebra Law or Rule A + 1 = 1 A in parallel with closed = "CLOSED" Annulment A + 0 = A A in parallel with open = "A" Identity A . 1 = A A in series with closed = "A" Identity A . 0 = 0 A in series with open = "OPEN" Annulment A + A = A A in parallel with A = "A" Indempotent A . A = A A in series with A = "A" Indempotent NOT A = A NOT NOT A Double Negation
- 8. Fredrick Kendrick Electrical Engineering Dept. ET1220 (double negative) = "A" A + A = 1 A in parallel with not A = "CLOSED" Complement A . A = 0 A in series with not A = "OPEN" Complement A+B = B+A A in parallel with B = B in parallel with A Commutative A.B = B.A A in series with B = B in series with A Commutative A+B = A.B invert and replace OR with AND de Morgan’s Theorem A.B = A+B invert and replace AND with OR de Morgan’s Theorem The basic Laws of Boolean Algebra that relate to the Commutative Law allowing a change in position for addition and multiplication, the Associative Law allowing the removal of brackets for addition and multiplication, as well as the distributive Law allowing the factoring of an expression, are the same as in ordinary algebra. Each of the Boolean Laws above are given with just a single or two variables, but the number of variables defined by a single law is not limited to this as there can be an infinite number of variables as inputs too the expression. These Boolean laws detailed above can be used to prove any given Boolean expression as well as for simplifying complicated digital circuits. A brief description of the various Laws of Boolean are given below with A representing a variable input. Description of the Laws of Boolean Algebra Annulment Law – A term AND´ed with a “0” equals 0 or OR´ed with a “1” will equal 1. o A . 0 = 0, A variable AND’ed with 0 is always equal to 0. o A + 1 = 1, A variable OR’ed with 1 is always equal to 1. Identity Law – A term OR´ed with a “0” or AND´ed with a “1” will always equal that term.
- 9. Fredrick Kendrick Electrical Engineering Dept. ET1220 o A + 0 = A, A variable OR’ed with 0 is always equal to the variable. o A . 1 = A, A variable AND’ed with 1 is always equal to the variable. Indempotent Law – An input AND´ed with itself or OR´ed with itself is equal to that input. o A + A = A, A variable OR’ed with itself is always equal to the variable. o A . A = A, A variable AND’ed with itself is always equal to the variable. Complement Law – A term AND´ed with its complement equals “0” and a term OR´ed with its complement equals “1”. o A . A = 0, A variable AND’ed with its complement is always equal to 0. o A + A = 1, A variable OR’ed with its complement is always equal to 1. Commutative Law – The order of application of two separate terms is not important. o A . B = B . A, The order in which two variables are AND’ed makes no difference. o A + B = B + A, The order in which two variables are OR’ed makes no difference. Double Negation Law – A term that is inverted twice is equal to the original term. o A = A, A double complement of a variable is always equal to the variable. de Morgan´s Theorem – There are two “de Morgan´s” rules or theorems, (1) Two separate terms NOR´ed together is the same as the two terms inverted (Complement) and AND´ed for example, A+B = A. B. (2) Two separate terms NAND´ed together is the same as the two terms inverted (Complement) and OR´ed for example, A.B = A +B. Other algebraic laws not detailed above include: Distributive Law – This law permits the multiplying or factoring out of an expression. Absorptive Law – This law enables a reduction in a complicated expression to a simpler one by absorbing like terms. Associative Law – This law allows the removal of brackets from an expression and regrouping of the variables.
- 10. Fredrick Kendrick Electrical Engineering Dept. ET1220 Algebra Functions Using the information above, simple 2-input AND, OR and NOT Gates can be represented by 16 possible functions as shown in the following table. Function Description Expression 1. NULL 0 2. IDENTITY 1 3. Input A A 4. Input B B 5. NOT A A 6. NOT B B 7. A AND B (AND) A . B 8. A AND NOT B A . B 9. NOT A AND B A . B 10. NOT A AND NOT B (NAND) A . B 11. A OR B (OR) A + B 12. A OR NOT B A + B 13. NOT A OR B A + B 14. NOT OR (NOR) A + B 15. Exclusive-OR A.B + A.B 16. Exclusive-NOR A.B + A.B Laws of Boolean Algebra Example No1 Using the above laws, simplify the following expression: (A + B)(A + C) Q = (A + B)(A + C) AA + AC + AB + BC - Distributive law A + AC + AB + BC - Identity AND law (A.A = A)
- 11. Fredrick Kendrick Electrical Engineering Dept. ET1220 A(1 + C) + AB + BC – Distributive law A.1 + AB + BC - Identity OR law (1 + C = 1) A(1 + B) + BC - Distributive law A.1 + BC - Identity OR law (1 + B = 1) Q = A + BC - Identity AND law (A.1 = A) Then the expression: (A + B)(A + C) can be simplified to A + BC Instructor: Don Heller. ITT Technical Indianapolis.