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Boolean Algebra

Boolean Algebra. Summary. Digital Circuits (Chips/ICs/Transistors) Basic Logic gates Boolean Algebra, Basic Rules and Identities Logic Simplification using Boolean Algebra De Morgan’s Theorem and its Application in Logic Simplification Karnaugh Maps – Logic simplification using K’Maps

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Boolean Algebra

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  1. Boolean Algebra

  2. Summary • Digital Circuits (Chips/ICs/Transistors) • Basic Logic gates • Boolean Algebra, Basic Rules and Identities • Logic Simplification using Boolean Algebra • De Morgan’s Theorem and its Application in Logic Simplification • Karnaugh Maps – Logic simplification using K’Maps • Implicants

  3. Chips/ICs • Our world is full of integrated circuits (ICs) • We can found ICs starting from Microprocessor in our computer to almost every modern electrical device such as Car, TV, CD Player, Cell Phone, Electric oven, washing machine etc. • Made from different electrical components such as transistors, resistors, capacitors and diodes

  4. Gordon E. Moores’s Laws • Transistors per an IC: Doubling of the number of transistors on integrated circuits every two years (at least for 1 more decade) • Cost per transistor: As the size of transistors has decreased, the cost per transistor has decreased as well • Computing performance per unit cost: As the size of transistors shrinks, the speed at which they operate increases

  5. Transistor count is the most common measure of chip complexity.

  6. Boolean algebra • There are only two possible values for any quantity and for any arithmetic operation 1 or 0 • It does not matter how many or few terms we add together, either.

  7. Addition in Boolean Algebra • Is not same as real-number algebra

  8. Multiplication in Boolean Algebra • Is same as in real-number algebra. Anything multiplied by 0 is 0, and anything multiplied by 1 remains unchanged

  9. Logic Gates • A logic gate is an elementary building block of a digital circuit. • There are AND, OR, NOT, NAND, NOR, EXOR and EXNOR gates. • Most logic gates have two inputs and one output. • At any given moment, every terminal is in one of the two binary conditions low (0) or high (1), represented by different voltage levels. • The logic state of a terminal can, and generally does, change often, as the circuit processes data. • In most logic gates, the low state is approximately zero volts (0 V), while the high state is approximately five volts positive (+5 V).

  10. Boolean Addition corresponds to the logical function of an "OR" gate

  11. Boolean addition corresponds to the logical function of an “AND" gate

  12. Boolean compliment corresponds to the logical function of a “NOT" gate

  13. Boolean Identities • The sum of anything and zero is the same as the original "anything." • This identity is no different from its real-number algebraic equivalent

  14. Boolean Identities • The sum of anything and one is one • Different from normal algebra

  15. Boolean Identities • Adding A and A together • Is same as connecting both inputs of an OR gate to each other and activating them with the same signal

  16. Boolean Identities • The sum of a variable and its complement is 1

  17. Boolean Identities • Just as there are four Boolean additive identities (A+0, A+1, A+A, and A+A'), so there are also four multiplicative identities: Ax0, Ax1, AxA, and AxA'. Of these, the first two are no different from their equivalent expressions in regular algebra:

  18. Boolean Identities • The third multiplicative identity: The product of a Boolean quantity and itself is the original quantity, since 0 x 0 = 0 and 1 x 1 = 1

  19. Boolean Identities • The fourth multiplicative identity: The product of a variable and its complement is 0

  20. Boolean Identities (Summary)

  21. Boolean Identities • Double complement: a variable inverted twice. Complementing a variable twice (or any even number of times) results in the original Boolean value.

  22. Laws of Boolean Algebra • The commutative law/property tells that, we can reverse the order of variables that are either added together or multiplied together without changing the truth of the expression

  23. Laws of Boolean Algebra • Associative law tells that, we can associate groups of added or multiplied variables together with parentheses without altering the truth of the equations

  24. Laws of Boolean Algebra • Distributive property: The Boolean expression formed by the product of a sum, and in reverse shows how terms may be factored out of Boolean sums-of-products

  25. Basic Boolean Algebraic properties • Commutative, Associative, and Distributive

  26. Boolean Rules • Boolean algebra finds its most practical use in the simplification of logic circuits. • If we translate a logic circuit's function into symbolic (Boolean) form, and apply certain algebraic rules to the resulting equation to reduce the number of terms and/or arithmetic operations, the simplified equation may be translated back into circuit form for a logic circuit performing the same function with fewer components. • If a equivalent function may be achieved with fewer components, the result will be increased reliability and decreased cost of manufacture.

  27. Boolean Rules • This rule may be proven symbolically by factoring an "A" out of the two terms, then applying the rules of A + 1 = 1 and 1A = A to achieve the final result:

  28. Boolean Rules

  29. Boolean Rules • Proving using truth table

  30. Boolean Rules • Simplification of a product-of-sums expression

  31. Boolean Rules (Summary)

  32. DeMorgan's Theorems

  33. DeMorgan's Theorems • Reducing the expression (A + (BC)')' to A’BC using DeMorgan's Theorems

  34. DeMorgan's Theorems • Reducing the expression (A + (BC)')' to A’BC using DeMorgan's Theorems

  35. DeMorgan's Theorems • Maintaining the grouping implied by the complementation bars for the expression is crucial to obtaining the correct answer

  36. DeMorgan's Theorems • Applying the principles of DeMorgan's theorems to the simplification of a gate circuit • Label the outputs of the first NOR gate and the NAND gate • Finally, write an expression (or pair of expressions) for the last NOR gate

  37. DeMorgan's Theorems • Reduce the expression using the identities, properties, rules, and theorems (DeMorgan's) of Boolean algebra

  38. DeMorgan's Theorems (Review) • DeMorgan's Theorems describe the equivalence between gates with inverted inputs and gates with inverted outputs. Simply put, a NAND gate is equivalent to a Negative-OR gate, and a NOR gate is equivalent to a Negative-AND gate. • When "breaking" a complementation bar in a Boolean expression, the operation directly underneath the break (addition or multiplication) reverses, and the broken bar pieces remain over the respective terms. • It is often easier to approach a problem by breaking the longest (uppermost) bar before breaking any bars under it. You must never attempt to break two bars in one step! • Complementation bars function as grouping symbols. Therefore, when a bar is broken, the terms underneath it must remain grouped. Parentheses may be placed around these grouped terms as a help to avoid changing precedence.

  39. Karnaugh Maps • Applying Boolean algebra can be awkward in order to simplify expressions • It is laborious and requires remembering all the laws • The Karnaugh map provides a simple and straight-forward method of minimizing Boolean expressions • With the Karnaugh map Boolean expressions having up to four and even six variables can be simplified. • Karnaugh map provides a pictorial method of grouping together expressions with common factors and therefore eliminating unwanted variables. • Karnaugh map can also be described as a truth table.

  40. Karnaugh Maps • Minterm: (Standard product or canonic product term) such as AB’CD or A’BCD’ etc. where each variable used once and once only. • Maxterm: (Standard sum or canonical sum term) such as (A+B’+C+D) or (A’+B+C+D’) where each variable used once and once only • Sum of products: (Minterm canonic form or canonic sum function f(A,B,C,D)=AB’CD+A’BCD’+A’BC’D • Product of sums: (Maxterm canonic form or canonic product function f(A,B,C,D)=(A+B’+C+D) (A’+B+C+D’)(A’+B+C’+D) • Adjacent Cells: If two occupied cells of a Karnaugh are adjacent, horizontally, vertically (but not diagonally) then one variable is redundant. Adjacent cells differ in the value of only one variable. (Rule of adjacency - can knock off one variable as A+A’=1)

  41. Karnaugh Maps • Combining all adjacent 1’s more than once doesn’t matter unless no 1 is left out, as A + A = A and A.A = A • Physical, Logical adjacency

  42. Karnaugh Maps • The correspondence between the Karnaugh map and the truth table (two variable) • The values inside the squares are copied from the output column of the truth table, therefore there is one square in the map for every row in the truth table

  43. Karnaugh Maps • Consider the following map. The function plotted is: Z = f(A,B) = A B’+ AB • Referring to the map the two 1’s are grouped together. The variable B has its true and false form within the group. Eliminate B leaving only A which only has its true form • Using algebric simplication: Z = A + AB Z = A( + B) Z = A

  44. Karnaugh Maps • Consider the expression z = f(A,B) = A’B’+AB’+A’B plotted on Karnaugh map: • The first group labeled I, consists of two 1s which correspond to A = 0, B = 0 and A = 1, B = 0. Put in another way, all squares in this example that correspond to the area of the map where B = 0 contains 1s, independent of the value of A. So when B = 0 the output is 1. The expression of the output will contain the term • For group labeled II corresponds to the area of the map where A = 0. The group can therefore be defined as . This implies that when A = 0 the output is 1. The output is therefore 1 whenever B = 0 and A = 0Hence the simplified answer is Z = A’ + B’

  45. Karnaugh Maps • Given the truth table The Boolean algebraic expression is m = a'bc + ab'c + abc' + abc. Minimization is done as follows. m = a'bc + abc + ab'c + abc + abc' + abc = (a' + a)bc + a(b' + b)c + ab(c' + c) = bc + ac + ab bc + ac + ab

  46. Karnaugh Maps • The Karnaugh map for 4 variables q = a'bc'd + a'bcd + abc'd' + abc'd + abcd + abcd' + ab'cd + ab'cd' q = bd + ac + ab This expression requires 3 2-input and gates and 1 3-input or gate. RULE: Minimization is achieved by drawing the smallest possible number of circles, each containing the largest possible number of 1s.

  47. Karnaugh Maps • Imlpicant: Each of the terms i.e product terms that are combined to become sum of products later on are called implicants • Prime Imlpicant: Largest possible group of values. For that group we can not find larger group An implicant can get submerged in to a prime implicant. • Essential Prime Imlpicant: At least one 1 or cell, which not been covered in any other group should be covered • Non Essential Prime Imlpicant: One way of combining 1s which is not covered otherwise as a an essential prime implicant

  48. Gordon E. Moore • Each year computer chips become more powerful yet cheaper than the year before. Gordon Moore, one of the early integrated circuit pioneers and founders of Intel once said, • "If the auto industry advanced as rapidly as the semiconductor industry, a Rolls Royce would get a half a million miles per gallon, and it would be cheaper to throw it away than to park it.”

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