c and d represent the 1st and 2nd bits of a binary number N. In the above truth table, the minterms will be m 2, m 5, m 6 and m. ) (c) Express F′ as a minterm expansion. 90% (10 ratings) for this solution. Question: Using kmap to find the SOP format for the Boolean function below, Z, Z = (A + B' + C)(A' +B+ C') choose the closet answer from the multiple choice hint: Z expressed in POS format so find Maxterm expansion using truth table. For each row with a 1 in the final column, the corresponding product to add to the minterm expansion is obtained as: COE/EE 243 Digital Logic Session 9; Page 5/9 Spring 2003 x y x y 3. are those nos. Finally map minterm expansion of Z into K-map O Z = AB +AC + A'B O Z = B'C' + A'C +. . 29 Find both the minterm expansion and max-term expansion for the following func- tions, using algebraic manipulations: (a) B, C, D) = AB +A'CD . g. A and B represent the first id second bits of an unsigned binary number N. (c) (1 Points) Write down an expression for F ′ in terms of minterms. This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. (d) (1 Points) Using summation ( Σ m j ) and product (I M j ) notation, show that the maxterm expansion of F has the same indices as the minterm expansion of F ′. Note that the function is unchanged if the exclusive OR is replaced with inclusive OR. Question: Due Date: Friday, 3/31/2023 Show your complete work for each problem - Write down the algebraic expression for minterm and maxterms for the following cases. (a) Find the minterm expansion for F. Express Fas a minterm expansion in boolean expression. I have a question regarding the process of finding minterms. 1. How to express a digital logic function in minterm (sum of products) form. 1. 2: Minterm. This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Using scientific means, determine the minterm expansion for the given function: F = X'Y + XY' + Y'Z'. Find step-by-step solutions and answers to Exercise 27 from Fundamentals of Logic Design - 9781111781392, as well as thousands of textbooks so you can move forward with confidence. Therefore, the minterm expansion for is . 2. ) Show transcribed image text. Given the following truth table: A B C F 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 a) Write down the minterm expansion for the function F b. Give the minterm and maxterm expansion for each of the following logic. and Mi drops out of the expansion; however, Mi is present if ai = 0. Equation (4-1) can be rewritten in terms of m-notation as: This can be further abbreviated by listing only the decimal subscripts in the. 1 Answer Sorted by: 0 ab+ac' ab (c+c')+a (b+b')c' abc+abc'+abc'+ab'c' abc+abc'+ab'c' use a truth table with 3 variables a,b,c find followings abc = 111 - 7 abc' =. . General Minterm and Maxterm Expansions list maxterms not present in F minterm nos. Step-by-step solution. A minterm is a Boolean expression resulting in 1 for the output of a single cell, and 0s for all other cells in a Karnaugh map, or truth table. Maxterm to Minterm conversion: rewrite maxterm shorthand using minterm shorthand replace maxterm indices with the indices not already used E. A and B represent the first and second bits of a binary number N. . For each row with a 1 in the final column, the corresponding product to add to the minterm expansion is obtained as:Minterms. From the Digital Design course. Computer Science 3. Simplified Boolean Function (Minimum SOP): Let's simplify the. Minterm expansion of Y: Y = A'B' + A'B c. b) F as a max term expansion is . MINTERM Meaning. 5. 문제를 많이 풀어보시길 바랍니다. Question: 2. Minterms and Maxterms are important parts of Boolean algebra. Minterm. ). 민텀을가진다. Consider the following function, which is stated in minterm expansion form: F(W,X,Y,Z)=Σm(0,4,5,7,12)+∑d(2,9,11,13,15) Using Karnaugh maps, a. Specify ai and bi, F1=Σaimi and F2=Σbimi (b) Find the minterm expansion of (F1+F2)′ use ∑i=0i=7mi=1 and X′+X=1 (c) Find the maxterm expansion of F1∗F2, obtain mintern and then convert it to maxterm (d) Find. Maxterm to Minterm conversion: rewrite maxterm shorthand using minterm shorthand replace maxterm indices with the indices not already used E. Prove your answer by using the general form of the minterm expansion. Given expression is. With n variables,. SOP representation is the sum of minterm where minterm is the function variable combination for which the output is high (1). Electrical Engineering questions and answers. (a) Write the minterm expansion for F. , the value of D is such that A, B, C, and D always contain an odd number of 1’s. . To, prove the answer, the minterms can be expanded using. 2，Provide the minterm expansion and the maxterm expansion of F’s complement, respectively. (Use m-notation. A and B represent the first and second bits of a binary number N 1. ) Keep the first 1, and x amount of 0's before the first one. Express F as a minterm expansion in boolean expression. Given F1= Σ m(0, 4, 5, 6) and F2= Σ m(0, 3, 6, 7) find the minterm expression for F1+F2. Represent the heads or tails status of each coin by a logical variable (A for the first coin, B for the second coin, and C for the third) where the logical variable is 1 for heads and 0. 100% (1 rating) Transcribed image text: For the truth table given below, find The Standard SOP expression, or the minterm expansion of f The Standard POS expression, or the maxterm expansion of f The. 1-minterms = minterms for which the function F = 1. Prove your answer by using the general form of. F'=M0M1M2. This product is not arithmetical. 6) F (A,B,C)=m3+m4+m5+m6+m7. logicly. Given: F (a, b, c) = abc’ + b’. Keep your answer in the minterm expansion form i. Design a binary adder which takes three inputs: A,B and C ( C is the carry bit, A and B are the two binary numbers), there are two outputs S and X ( S is the sum and X is carry bit to next column) and the table is given as below: a) Write down the minterm expansion for S (in terms of A,B and C ) b) Write down the minterm expansion. m0+m1+m6+m7. 2. EE/CompE 243 Digital Logic Session 10; Page 2/4 Spring 2003 (c) F A B C. Z' +. ) (c) Express F as a minterm expansion. Step 1. This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. For example, , ′ and ′ are 3. F = Σ m (0,1,4,5,6) (b) Express F as a maxterm expansion (Use M -notation): F = ∏ = ∏ F = Π M ( 0,2,3,8,9,12,13,14,15 )(a) Find the minterm expansion for F (both decimal and algebraic form) (b) Find the maxterm expansion for F (both decimal and algebraic form) and 5. 8 Express your answers inMinterm expansion = ∑ m (1, 2, 4) ext{Minterm expansion}=sum m(1,2,4) Minterm expansion = ∑ m (1, 2, 4) The minterm expansion will consist of a sum of 3 3 3 products when the truth table contains 3 3 3 ones in the last column. 20 A bank vault has three locks with a different key for each lock. C and D represent the first and second bits of a binary number N. See Answer. For this truth table the minterm of row (a) is The minterm for row (b) is • The expression y= (as a minterm expansion) e f y 0 0 0 0 1 1 (a) a 1 0 0 1 1 1 (b) Show. We consider in this chapter real random variables (i. If A = 0, the output Z is the equivalence of B OR C. ) (d) Express F' as a maxterm expansion. Write the complete minimized Boolean expression in sum-of-products form. "This form is composed of two groups of three. Expert Answer. (Use M-notation. Ask AI. (a) Consider the function. Without using a truth table, find. (b) Express f as maxterm expansion (use M-notation). It is often referred to as minterm expansion or standard sum of products. Show transcribed image text. Write a logic function F(A, B, C) which is 1 if and only if exactly one of the coins is heads after a toss of the coins. (Use M-notation. ) (d) Express F′ as a maxterm expansion. Sum of product form is a form of expression in Boolean algebra in which different product terms of inputs are being summed together. This expression can then be converted to decimal notation. Minterm expansion = ∑ m (0, 1, 2, 4, 8) ext{Minterm expansion}=sum m(0,1,2,4,8) Minterm expansion = ∑ m (0, 1, 2, 4, 8) The minterm expansion will consist of a sum of n n n products when the truth table contains n n n ones in the last column. Each chair may be occupied (1) or empty (0). 100% (42 ratings) for this solution. Find the minterm and maxterm expansion for f in decimal and algebraic form. Write the complete minimized Boolean expression in sum-of-products form. (2 points) An 8×1 MUX has control (or, select) variables A, B, C. Sum of product form is a form of expression in Boolean algebra in which different product terms of inputs are being summed together. 2. Solution for Applying the different laws and identities in Boolean algebra, determine the minterm expansion of the given Boolean expression: F(W,X,Y,Z) = X’Z +…Definition 12. Question: Q3. Use the minterm expansion principle to create a circuit that uses only AND, OR, and NOT gates. The output is to be 1 only if the proc x N2 is less than or equal to 3. The output of the maxterm functions. (b) Write the minterm expansion for Z in decimal form and in terms of A,B,C. Fill in the truth table below with the minterm expansion for f= m (1,3,5,6,7) and the maxterm expansion for g= II M (0,1,2,3,4,6) 09 ABCf 0 0 0 0 0 1 0 1 0 011 1 0 0 101 110 1 1 1 Use a 3-variable Karnaugh map to determine a Boolean expression in terms of A, B, and Cfor function f as. 4. A = AB (C + C’) + AB' (C + C’) = ABC + ABC’+ AB’C + AB’C’. a) As a minterm expansion b) A. Most common MINTERM abbreviation full form updated in November 2023. (Use M-notation. Let a = 1. ) (c) Express F as a minterm expansion. (a) Complete the truth table. Equivalence of B and C is defined as: (B⊕C)′ 1. Who are the experts? Experts are tested by Chegg as specialists in their subject area. In this we learn about What are Minterms, Different Procedures for Minterm expansion of expression, Shorthand. Write a logic function F (A, B, C) which is 1 iff. 100% (4 ratings) for this solution. Otherwise, the output is zero. Check out a sample Q&A here. See below for information about how to import your Full Adder, as well as for the steps needed to export Ripple-Carry Adder. 27 Given f (a, b, c) = a (b + c'). Q1) a) Write the functions F (x, y, z) and G (x, y, z), which are represented in Table below, in the form of minterm expansion (sum of the products) and maxterm expansion (product of the sums). (a) Find the truth table for Z. 2진. + b 0 2 0 Same adder works for both unsigned and signed numbers To negate a number, invert all bits and add 1 As slow as add in worst caseD, and D next to A. c. (use m-notation) (d) Express F' as a maxterm expansion. E. Note: If a truth table is given, and if the output is 1 then it corresponds to minterm and in case the output is 0 then it corresponds to 0. Minterm expansion = ∑ m (0, 1, 2, 4, 8) \text{Minterm expansion}=\sum m(0,1,2,4,8) Minterm expansion = ∑ m (0, 1, 2, 4, 8) The minterm expansion will consist of a sum of n n n products when the truth table contains n n n ones in the last column. Otherwise, F = 0. Step 1 of 5. A combinational switching circuit has four inputs and one output as shown. 3. Find the minterm expansion equation for F using Quine-McCluskey method and prime chart. ) (c) Express F′ as a minterm expansion. Computer Science questions and answers. Show transcribed image textO 1. A Boolean expression or map may have multiple. Computer Science questions and answers. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site∏M 0 3 5 6 so the minterm expansion is. For example, if A = 1, B = 0, C = 0, and D = 0, then Z = 1, but if A = 0, B = 1, C = 0, and D = 0, then Z = 0. (a) Write the minterm expansion for F. That is, the cell 111 corresponds to the minterm ABC above left. Write out the full Boolean expressions. Computer Science. Select \(m\) distinct subclasses and form Boolean combinations for each of these. . C. In this course, you will study the history of modern computing technology before learning about modern computer architecture and a number of its essential features, including instruction sets, processor arithmetic and control, the Von Neumann architecture, pipelining, memory management, storage, and other input/output topics. This function is still missing one variable, so. Show transcribed image text. Task 2: Four chairs are placed in a row: А B с D Each chair may be occupied (1) or empty (0). A and B represe bits of a binary number N,. Find the truth table for Z. Minterm from. Do 4 and 5. loading. Each group of three is a 'minterm'. 🔗. ie. (20 points) C. b. The output is 1 iff the input has three consecutive 0’s or three consecutive 1’s. 최소항 전개 (minterm expansion)는 정규형 중 논리 함수의 모든 변수들을 곱의 합 형태로 표현한 것이다. It takes three inputs: x and y, the two bits being added from this column of the addition; c_in, the carry bit into this column of the additionDiscuss. (a) Express F as a minterm expansion (Use m-notation): F = ∑ (b) Express F as a maxterm expansion (Use M-notation): F = ∏ (c) Express F’ as a minterm expansion (Use. It takes three inputs: x and y, the two bits being added from this column of the addition. Maxterm expansion of Y: Y = (A + B')(A' + B) a. Solution –. Given: f (a, b, c) = a (b + c’). Computer Science questions and answers. The minterm expansion for \(E \cap F\) has only those minterms in both sets. Task 1: Given that F(a, b, c) = ac' + a'c+b' a) Create a truth table for the function b) Express F as a minterm expansion (use m-notation) c) Express F as a maxterm expansion (use M-notation) d) Express F' as a minterm expansion (use m-notation) e) Express F' as a maxterm expansion (use M-notation) f) Build Logisim circuits for b) and c) and show that. ) [5 Points] Q2) redo Q1 using the following function [10 Points: 5 points each] F(a,b,c,d)=(a+b+d)(a′+c)(a′+b′+c′)(a+b+c′+d′) Hint: You will need to add the missing variables in each expression by XX′=0 then9. (use M-notation) Expert Solution. For each row with a 1 in the final column, the corresponding product to add to the minterm. Last updated on Nov 11, 2023 HTET Application date extended! Now, the last date to submit applications is 11th November 2023. (a) Minterm expansion of is: To expand the given into a standard sum of products (minterm expansion) then introduce the missing variables in each term as. Equivalence of B and C is defined as: (B⊕C)′ 1. Minterm expansion; No lecture on 10/10 (holiday) Lecture on 10/11 (Mon. Computer Science questions and answers. a. simplifying boolean expression in minterm. 1-minterms = minterms for which the function F = 1. What the expression minterm is intended to imply it that each of the groups of three in the expression takes on a value of 1 only for one of the eight possible combinations of X, Y and Z and their inverses. The circuit to implement this function. This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Valid excess-3 code : 0011 to 1100 Invalid excess-3 code : [0000 to 0010] and [1101 to 1111] 10. Let G = (V,E) be a graph. Express F as a minterm expansion (standard sum of product). Any Boolean function can be expressed as a sum (OR) of. (b) Find the maxterm expansion for F (both decimal and algebraic form). Using algebraic manipulations the required minterm expansion is to expand the given into a standard sum-of-products (minterm expansion) then introduce the missing variables. (Use m-notation. Design a combinational logic circuit which has one output Z and a 4. ) [5 Points] (b). Z = ∑ (c) Find the maxterm expansion for Z in both decimal form in terms of A, B, C. What is the minterm expansion of F? ii. For each row with a 1 in the final column, the corresponding product to add to the minterm expansion is obtained as:COE/EE 243 Digital Logic Session 9; Page 5/9 Spring 2003 x y x y 3. 27 Given f (a, b, c) = a (b + c). Using postulates and theorems of Boolean algebra, simplify the minterm expansion F to a form with as occurrences of each variable as possible. A Boolean function can be represented in the form of sum of minterms or the product of maxterms , which enable the designer to make a truth table more easily. Write POS from maxterm expansion F = (A+B +C)(A+B0 +C)(A0 +B +C) 4. (Use M-notation. 민텀 (Minterm) (최소항) 이때 곱의 항 (곱들의 합)을 민텀 (Minterm) 이라고 한다. GATE CS 2010. 1. 0-minterms = minterms for which the function F = 0. i. The input D is an odd-parity bit, i. Who are the experts? Experts have been vetted by Chegg as specialists in this subject. Thank You so much!Minterm expansion of f(P, Q, R) is m 2 + m 4 + m 6 + m 7. 1. c. Above right we see that the minterm A’BC’corresponds directly to the cell 010. Stuck at simplifying boolean expression. e. A switching circuit has four inputs as shown. Computer Science. (Use m-notation. d. 1. The given expression can be written as: F(A, B, C) = AB(C + C̅) + (A + A̅) BC̅ + A(B + B̅)C̅ Minterm. Express your answer in both decimal notation and algebraic formBuild a Full Adder (FA) as described in lecture, using the minterm expansion principle to create a circuit that uses only AND, OR, and NOT gates. (a)True (b)False?? The Sum of Product (SOP) canonical form is also known as ____?? (a)Minterm expansion (b)Maxterm expansion The 1’s complement of the. (use M-notation) (c) Express F' as a minterm expansion. See below for information about how to import your Full Adder, as well as for the steps needed to export Ripple-Carry Adder. d) F′ as a max term expansion is . Maxterm expansion: To find the maxterm expansion, we need to find the terms that are not present in the minterm expansion. b. Use the minterm expansion principle to create a circuit that uses only AND, OR, and NOT gates. A truth table for the function F(a, b, c) = ac' + a'c+b' can be created by evaluating the function for all possible combinations of inputs a, b, and c. are the same as minterm nos. Sum of Product is the abbreviated form of SOP. (Use M-notation. It is sometimes convenient to express a Boolean function in its sum of minterm form. 아래 예를 통해 이해해봅시다. During the first step of the method, all prime implicants of a function are. The point to review is that the address of the cell corresponds directly to the minterm being mapped. A combinational switching circuit has four inputs and one output as shown. 100% (42 ratings) for this solution. "Note that the sum has five bits to accommodate the possibility of a carry. What does minterm mean? Information and translations of minterm in the most comprehensive. 9, which involves an independent class of ten events. Express F as a minterm expansion in boolean expression. Example (PageIndex{12}) A hybrid approach. , F(A,B,C) = ΠM(0,2,4) F’(A,B,C) = ΠM(1,3,5,6,7) An equation can be proven valid by factoring the minterm expansions of each side and showing the expansions are the same Ex) a'c + b'c' + ab = a'b' + bc + ac' Left side, = a'c (b + b') + b'c' (a+ a') + ab (c + c') = a'bc + a'b'c + ab'c' + a'b'c' + abc + abc' 011 001 100 000 111 110. Slide 1 Boolean Algebra and Logic Simplification BY UNSA SHAKIR. For example, , ′ and ′ are 3. Note that the sum has five bits to accommodate the possibility of a carry. (a) Write the minterm expansion for F. Simplify further. 我们首先开始于定义极小项(minterm)为只由逻辑与和补运算符组成的 n 个变量的逻辑表达式。 例如，下列是极小项的例子: a b'c a' b c. not on the maxterm list for F Maxterm Expansion of F maxterm nos. Then you derive the input equations of the three FFs (six ones). Simplifying boolean expression from truth table. maxterm expansion) and draw the corresponding circuit. A minterm, denoted as mi, where 0 ≤ i < 2n, is a product (AND) of the n variables in which each variable is complemented if the value assigned to it is 0, and uncomplemented if it is 1. General Minterm and Maxterm Expansions list maxterms not present in F minterm nos. 5. of F minterm nos. Maxterm expansion of X: X = (A + B)(A + B') d. Minterm is the product of N distinct literals where each literal occurs exactly once. B. e. the minterm/maxterm expansions can be obtained from truth tables as described above (or can also obtain minterm expansion by using sum of products and by interoducing the missing variables in each term, X + X' = 1) or for maxterm expression by using X X' =0 and factoring. 90% (10 ratings) for this solution. Using theorems of Boolean algebra, simplify the minterm expansion of F to a form with as few occurrences of each as possible. Use of the minterm expansion for each of these Boolean combinations and the two propositions just illustrated shows that the class of Boolean combinations is independent. Expert Answer. Express F as a minterm expansion in boolean expression. Express F as a minterm expansion in m-notation. XY represents a binary number whose value equals the number of O's at the input. 4, General Minterm and Maxterm Expansions. 100% (27 ratings) for this solution. Example: Calculate the length change of a bronze bar (L = 5m, α = 18 ×10 -6 /°C), if the temperature rises from 25°C to 75°C. Q3. Give the minterm and maxterm expansion for each logic. Q3. I mostly need help with building the circuits for each. Computer Science questions and answers. The input D is an odd-parity bit, i. Question: 50 pts. Write out the full Boolean expressions. State a general rule for finding the expression for F1+F2 given the minterm expansions for F1 and F2. Trending now This is a popular solution! Step by step Solved in 2 steps. A minterm is a Boolean expression resulting in 1 for the output of a single cell, and 0 s for all other cells in a Karnaugh map, or truth table. 3 a Ми використовуємо шаблони, відображені на карті minterm, щоб допомогти в алгебраїчному розв'язанні для різних мінтермальних ймовірностей. 9th Edition. General Minterm and Maxterm Expansions general function of n variables A minterm is an expression regrouping the Boolean variables, complemented or not (a or not (a)), linked by logical ANDs and with a value of 1. me/tanmaysakpal11-----. The output of the minterm functions is 1. are the same as maxterm nos. The output is to be l only if the product Ni × N, İS (a) Find the minterm expansion for F (b). each maxterm has a value 0 for exactly one combination of values for variables a=b=c=0 then maxterm is a+b+c and its abbreviated form M0. Duplicate term has been crossed out, because. Minterm to maxterm " Use maxterms that aren™t in minterm expansion " F(A,B,C) = ∑m(1,3,5,6,7) = ∏M(0,2,4)! Maxterm to minterm " Use minterms that aren™t in maxterm expansion " F(A,B,C) = ∏M(0,2,4) = ∑m(1,3,5,6,7) ! Minterm of F to minterm of F’ " Use minterms that don™t appear " F(A,B,C) = ∑m(1,3,5,6,7) F’(A,B,C) = ∑m(0,2,4)!A switching circuit has four inputs as shown. 0-minterms = minterms for which the function F = 0. Maxterm is the sum of N distinct literals where each literals occurs exactly once. Choose a proper value for the don’t care term and provide the shortest possible expression of POS (product of sum) for F . If A=1, the output is the equivalence of B and C. In Sum Of Products (SOP), each term of the SOP expression is called a "minterm" because, say, an SOP expression is given as: F(X,Y,Z) = X'. For each remaining row, create a minterm as follows: For each variable that has a 1 in that row, write the name of the variable. (b) Find the maxterm expansions for Y and Z. Write the minterm expansion for Z in decimal form and in terms of A,B,C. Given the following maxterm expansion. Find the maxterm expansion of f2(a,b,c,d)= bc + ac’ + abd’ 7. Given F1 = ∑ m(0,2,5,7,9) and F1 = ∑ m(2, 3,4,7,8) find the minterm expression for F1+F2. Minterm • Minterm or Sum of Product (SOP) • For output 1 • We do the product (AND) of input (0 = inverse) • For output 0 • We ignore them • Then, we do sum (OR) of all product A’BC. )(3pts) loading. (17 points) Y = Not the question you’re looking for? Post any question and get expert help quickly. This circuit should not be built using the minterm expansion principle. First, we will write the minterm: Maxterm = A+B'. Otherwise, F = 0. 1) F 1 ( x, y) = x ⋅ y ′ + x ′ ⋅ y + x ⋅ y. Minterm expansion = ∑ m (1, 2, 4) ext{Minterm expansion}=sum m(1,2,4) Minterm expansion = ∑ m (1, 2, 4) The minterm expansion will consist of a sum of 3 3 3 products when the truth table contains 3 3 3 ones in the last column. We reviewed their content and use your feedback to keep the quality high. The function F can be expressed as: F(A,B,C) = (A'B'C) + (A'BC) + (AB'C) + (ABC')Question: Q1- Which of the following is a tautology? a) -pv-q b) (p^g) → (q vp) c) (p^g)^(-(pvg)) d) pv (-p^q) e) (p→q) → p Q2 - Given Karnaugh map (see figure), write simplified minterm expansion. me/tanmaysakpal11-----. d) The decimal integers which are not in the min-terms list are called max-term of the function. (a) Write the minterm expansion for F. A B |С| D 3. The general rule for finding the expression, is that all the terms present in either or will be present in . Find step-by-step solutions and answers to Exercise 10 from Fundamentals of Logic Design - 9781133628477, as well as thousands of textbooks so you can move forward with confidence. 0 × 29 , b = −1. Given a truth table for a Boolean function, the minterm (also known as standard product) expansion could be constructed by taking an OR of products, where each product represents an instance. Find step-by-step solutions and answers to Exercise 27 from Fundamentals of Logic Design - 9781133628477, as well as thousands of textbooks so you can move forward with confidence. Prove your answer by using the general form of the minterm expansion.