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Lecture 4: Boolean Algebra II Standard Forms Our purpose here is to consider some further Boolean algebraic structures which provide standardised ways of expressing logic functions and which later lead to non-algebraic methods of simplification including computer-based methods. Learning Outcomes: On completing this lecture, you will be able to: Identify the minterms and maxterms of a given logic function; Express a Boolean function in canonical sum-of-products form; Convert the canonical sum-of-products to a product-of-sums; Specify the total set of functions of N variables. 4.1 Standard Forms Consider a general Boolean function F of three variables A, B, C, in truth table form: Row 0 1 2 3 4 5 6 7 A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 F f0 f1 f2 f3 f4 f5 f6 f7 Minterm A’.B’.C’ A’.B’.C A’.B.C’ A’.B.C A.B’.C’ A.B’.C A.B.C’ A.B.C There are eight rows in the truth table and for each row we designate the value of the function by fi, ie each fi represents a 1 or 0 as appropriate. Note also that each row in effect corresponds to a three-literal product drawn from A, A’, B, B’, C, C’. The three-literal product term is called a minterm m5 A.B .C eg From the truth table specification we can express the function as follows: F f 0 .m0 f1 .m1 f 2 .m2 f 3 .m3 f 4 .m4 f 5 .m5 f 6 .m6 f 7 .m7 This particular formulation is referred to as the canonical sum-of-products or sum-ofminterms. Example 4.1: F A B.C A 1 OR B 0 AND C 1 Consider the function We can state in words: F 1 if statement we can generate the truth table: 4-1 and from this A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 F 0 1 0 0 1 1 1 1 Thus F can be written F 0. AB C 1. AB C 0. ABC 0. ABC 1. AB C 1. AB C 1. ABC 1. ABC m1 m4 m5 m6 m7 or more compactly F 1,4,5,6,7 4.2 Product of Sums1 Consider as before F m1 m4 m5 m6 m7 The complement of F will comprise those minterms which are NOT included in F: F m0 m2 m3 AB C ABC ABC Complementing this to get back to F, we get F [ AB C ABC ABC ] AB C ABC ABC A B C A B C A B C This product-of-sums formulation is based on the following so-called maxterms: A B C M 0 A B C M 2 A B C M 3 The maxterm index can be found by mapping an uncomplemented variable to 0 and a complemented variable to 1 the opposite convention to indexing a minterm. The function F can thus be written 1 This section may be omitted; it will not be examined. 4-2 F M 0M 2M 3 0,2,3 Alternatively, the product-of-minterms can be read from truth table entries with a 0: Row 0 1 2 3 4 5 6 7 A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 F 0 1 0 0 1 1 1 1 Maxterm A+B+C A+B’+C A+B’+C’ Note that, between the canonical sum-of-products and the product-of-sums, the full set of indices in the range 0 to 2N-1 is used 4.3 All Possible Two-Variable Functions Consider a two-input logic circuit described by a two-variable Boolean equation F(A,B). We ask: how many different such logic circuits are there? Or equivalently: how many different two-variable logic circuits are there? Noting that a two-variable Boolean equation may be written in canonical sum-of-products form F f 0 . AB f1 . AB f 2 . AB f 3 . AB the question is essentially: in how many different ways can we assign values to the set of coefficients f0, f1, f2, f3? Since each f i 0 or 1 the number of such functions is 24 = 16. This set of functions can be specified by means of a “function truth table” based on the f i coefficients: Function f0 f1 f2 f3 F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 4-3 Logic Operation F=AB F=A F=B F=A+B F=(A+B)’ F=B’ F=A’ F=(AB)’ Some of the more familiar logic functions such as AND, OR, NAND, NOR, etc are indicated. The important point, however, is that the table lists all possible two-variable logic functions. A more general question then would be: for a logic function/circuit of N variables, how many functions/circuits would there be? Again, the first step is to ask: how many fi coefficients will there be? Number f i 2 N 4.4 and Number functions 2 Number( fi ) 2 2 N The Exclusive-OR Function One of the more useful of the two-variable Boolean function is the so-called Exclusive-OR (XOR) function, denoted by the mathematical symbol ⊕ , and its complement, the ̅̅̅ . These functions are defined by the following Exclusive-NOR (XNOR) function, denoted ⊕ truth table and associated Boolean equations: A 0 0 1 1 B 0 1 0 1 𝑨⨁𝑩 0 1 1 0 𝑨⨁𝑩 1 0 0 1 𝐴 ⨁𝐵 = 𝐴. 𝐵′ + 𝐴′ . 𝐵 𝐴 ⨁ 𝐵 = 𝐴. 𝐵 + 𝐴′ . 𝐵′ We note that the XOR function corresponds to the regular OR function but excluding the possibility of both variables being 1 yielding a 1 output. Applied to three variables, A, B, C, the XOR function is given 𝐴 ⨁𝐵⨁𝐶 where it may readily be shown 𝐴⨁(𝐵⨁𝐶) = (𝐴⨁𝐵)⨁𝐶 = 𝐴⨁𝐵⨁𝐶 Although it is not a primitive gate in the sense that NAND or NOR gates are regarded as primitive, the XOR gate is given its own logic symbol A B X The most efficient way of generating the XOR function from primitive gates is 4-4 A X B The ouput X above is seen to be given by 𝑋 = {[𝐴 (𝐴 𝐵)′ ]′ [𝐵 (𝐴 𝐵)′ ]′ }′ = [𝐴 (𝐴𝐵)′ ] + [𝐵 (𝐴𝐵)′ ] = 𝐴 (𝐴′ + 𝐵′ ) + 𝐵 (𝐴′ + 𝐵′ ) which is indeed the Exclusive-OR function. = 𝐴 𝐵′ + 𝐴′ 𝐵 As will be seen later, both the XOR and XNOR functions are particularly useful in the implementation of arithmetic logic circuits. For example, the XNOR function indicates when two binary digits are equal as may be verified by referring to the truth table. Thus the following logic circuit can be used to indicate when two four-bit binary numbers are equal: a3 b3 a2 b2 E a1 b1 a0 b0 4.4 Conclusion In this lecture, we have developed two standard formulations of Boolean functions; the canonical sum-of-products/minterms and the canonical product-of-sums/maxterms. We have also explored the finite nature of Boolean functions, given a specific number of variables. We now proceed to employ this knowledge in developing further simplification methods. 4-5