That is true for only one combination of inputs. The cookies is used to store the user consent for the cookies in the category "Necessary". Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. Distributive Property; FOIL method; Difference of Squares; Analyze a Boolean expression: P and not Q P && (Q || R) Logic Circuits Minimal SOP form can be made using Boolean algebraic theorems but it is very easily made using Karnaugh map (K-map). Example of conversion for the above function in minimal SOP form is given below. How can you obtain SOP and POS (Boolean Logic) given the truth table? Through applying the laws, the function becomes easy to solve. Repeat the process for the 1 in the last line of the truth table. Feedback and suggestions are welcome so that dCode offers the best 'Boolean Minterms and Maxterms' tool for free! The copy-paste of the page "Boolean Minterms and Maxterms" or any of its results, is allowed as long as you cite dCode! Lets take an example of the function given above in canonical form. Simplify the following Boolean function in SOP form F(X,Y,Z) = m(1,4,5,6,7) Points to Remember. Don't Cares: Comma separated list of numbers. This expression is still in Sum of Product form but it is non-canonical or non-standardized form. Let's consider the numbers we need to find as x and y. The initial question concerning SOP and POS is how you can obtain it from a simple K-map or truth table of few inputs Boolean logic? The product of inputs is Boolean logical AND whereas the sum or addition is Boolean logical OR. Likewise, the OR function gives the logical sum of Boolean addition. In expression, it is represented by 4 x 5 = 20. The Sum-of-Products (SOP) expresses the combinational logic as two or more products are summed together. $$(\overline{A}+B+C)$$ Likewise, a Boolean sum-of-product (SOP) expression can be constructed from a given truth table. The Main Difference between Active and Passive Components, How to Calculate the Value of Resistor for LEDs & Different Types of LED Circuits, 15+ Must Have Android Apps for Electrical & Electronics Engineers & Students, Constant Current Diode Working, Construction, and Applications, You may also read: Negative Feedback and Negative Feedback Amplifier Systems, You may also read: Counter and Types of Electronic Counters. Disable your Adblocker and refresh your web page . Write the input variable if it is 1, and write the complement if the variable value is 0. A.1 = A. The Sum of Product (SOP) expression comes from the fact that two or more products (AND) are summed (OR) together. it is commercially beneficial because of its compact size, fast speed, and low fabrication cost. In mathematics, the number or quantity obtained by multiplying two (or more) numbers together is called theproduct. An Adder is a digital logic circuit in electronics that performs the operation of additions of two number. Why input combinations producing output '1' constitute a standard SOP expression and those producing output '0' constitute a standard POS expression? The achieved expression is the minimal product of sum form. These max terms are M, The Max terms are the complement of minterms. [other concept Minterms, A.B+AB]. We also use third-party cookies that help us analyze and understand how you use this website. Any 4 combinations of A, B, C, D, A, B, C & D represents the place values of 0 to 15 to address the cells of table in KMAP solver. (A+B) term is missing C input so we will add (CC) with it. This logical sum is known commonly as Boolean addition as anORfunction produces the summed term of two or more input variables, or constants. Therefore, when the products are OR'ed together, a one appears in the . For example, the multiplication of 4 by 5 is represented by 4 x 5 producing a product of 20. The non-standard forms can be converted to the sum-of-product form. OR the AND terms to obtain the output function. Feel free to contact us at your convenience! The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". A site about programming languages, Technical topics And for codes too. For example, the maxterm for the first row of a two-input truth table is ( A + B) because ( A + B) is FALSE when A = 0, B = 0. SOP is the default. When two or more sum terms are multiplied by a Boolean OR operation, the resultant output expression will be in the . Now narrow the founded expression down to minimal Product of Sume or POS form. I have tried to be sure that it is accurate but I can not guarantee it. Every two adjacent squares of the k-map have a difference of 1-bit including the corners. Replace the value of y in the equation x.y=44. The open-source game engine youve been waiting for: Godot (Ep. Output F is equal to zero but B is equal to 1. Example 3 In this example, we want to return the total of a particular item sold by a given region. 3 inputs have 8 different combinations. How to react to a students panic attack in an oral exam? Here the product in Boolean algebra is the logical AND, and the sum is the logical OR. Draw a logic gate diagram for the expression. The minterms and maxterms are two ways to see the same logical Boolean expression either with its 0 or with its 1 logic. Sum-of-Product form is a Boolean Algebra expression in which different "product" terms from inputs are "summed" together. The full adder (FA) circuit has three inputs: A, B and Cin, which add three input binary digits and generate two binary outputs i.e. We can also express it into canonical form as below Maxterm A sum term containing all the input variables of the function in either complemented or uncomplemented form is called a maxterm. Lets take the above-given function as example. Conversion from minimal or any sort of non-canonical form to canonical form is very simple. B is equal to 0 in the truth table when output F is equal to 1. It is also known as Product of Max term or Canonical conjunctive normal form (CCNF). Karnaugh map can produce Sum of product (SOP) or product of Sum (POS) expression considering which of the two (0,1) outputs are being grouped in it. It is used for logic tasks such as logic algebra and electronic circuits. Aproduct termcan have one or two independant variables, such asAandB, or it can have one or two fixed constants, again0and1. Maxterm means the term or expression that is true for a maximum number of input combinations or that is false for only one combination of inputs. But opting out of some of these cookies may affect your browsing experience. From the source of Wikipedia: Unary operations, Logical true, Logical false, Logical identity, Logical negation, Binary operations, Logical conjunction (AND), Logical disjunction (OR), Logical implication. The minimization can be carried out two-level or multi-level. In Binary Form Consider the following product of sum expression: Q = (A + B + C) (A + B + C) (A + B + C) For example, F(A, B) function has possible minterms of AB, AB, AB, and AB. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. In the following example, a non-standard Boolean expression is converted to sum-of-product (SOP). 3 inputs have 8 different combinations so it will have 8 maxterms. From the given sum-of-products expression following logic diagram is created. Please Whitelist Our Website by Allowing Ads, Sum Of Product (SOP) & Product Of Sum (POS), Product Of Sum (POS) & Sum Of Product (SOP), Canonical SOP expression is represented by summation sign, Which means that the function is true for the min terms, Conversion from Canonical SOP to Canonical POS, The remaining terms of this function are maxterms for which output is false. For example, the combinations ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD & ABCD represents the binary values of 0000, 0001, 0010, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110 & 1111 respectively. Boolean algebra is one such sub-branch of algebra. We Rely on Advertising to Help Fund Our Site and Provide Free Information. Your email address will not be published. Sum of the Products (SOP) Product of the Sums (POS) Standardization makes the evaluation, simplification, and implementation of Boolean expressions more systematic and easier. Asking for help, clarification, or responding to other answers. The terms in the sum-of-product expressions produce a logic 1 for that specific input state of terms. You may also read: Digital Synchronous Counter Types, Working & Applications. A truth table is a two-dimensional array with columns. 4 Variables Karnaugh's Map often known as 4 variables K-Map. getcalc.com's 3 Variables K-map solver, table & work with steps to find the Sum of Products (SOP) or to minimize the given logical (Boolean) expressions formed by A, B & C based on the laws & theorems of AND, OR & NOT gates in digital electronics. In Product of Sum each term is combined with AND operation and within each term, every variable combined with OR operation. Table of Contents Step 2: Use logical OR (i.e., '+') operators between all the minterms to get the expression in standard or canonical Sum of Product (SOP) form. For example, A = 0, or A = 1 whereas a Boolean constant which can also be in the form of a 1 or 0, is a fixed value and therefore cannot change. Get the free "Product/Sum Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Karnaugh Map (Kmap solver) calculator - group the terms, Sum of product, Product of sum, shows the simplified Boolean equation 2,3,4,5,6,7,8 Variable using Karnaugh Map (Kmap solver), step-by-step online. Tool for calculating Minterms (canonical disjunctive normal form) and Maxterms (canonical conjunctive normal form) from a truth table of a unknown Boolean expression. Necessary cookies are absolutely essential for the website to function properly. The third term: $$AB\overline{C}$$ For example, the above sum-of-product term can be simplified to:Q=A. Row 4: 1000, 1001, 1011, 1010. A.0 = 0, AND operation of a term with its complement is always equal to 0 i.e. Select the number of variables, then choose SOP (Sum of Products) or POS (Product of Sums) or Quine-McCluskey, and try some calculations. Example Lets say, we have a boolean function F defined on two variables A and B. Boolean expressions are simplified to build easy logic circuits. These Max terms will be used in POS expression as the product of these max terms. For example, the following Boolean function is a typical sum-of-product expression: However, Boolean functions can also be expressed in nonstandard sum of products forms like that shown below but they can be converted to a standard SOP form by expanding the expression. In a nutshell, the AND function gives the logical product of Boolean multiplication. This cookie is set by GDPR Cookie Consent plugin. We use cookies on our website to give you the most relevant experience by remembering your preferences and repeat visits. Boolean Algebra expression simplifier & solver. For example, the combinations ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD, ABCD & ABCD represents 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 & 15 respectively. A Boolean expression expressed as a sum of products (SOP) is also described as a disjunctive normal form (DNF). Output F is equal to zero but A is equal to 1. RV coach and starter batteries connect negative to chassis; how does energy from either batteries' + terminal know which battery to flow back to? a Maxterm is denoted as M. Y= M0+M1 Y = (A+B). note the truth table AB address. This truth table calculator will provide the truth table values for the given propositional logic formulas. Sum-of-Products and Product-of-Sums Expressions Digital Circuits PDF Version Question 1 Identify each of these logic gates by name, and complete their respective truth tables: Question 2 Identify each of these relay logic functions by name (AND, OR, NOR, etc.) Write the input variable if it is 1, and write the complement if the variable value is 0. A proposition is a set of declarative statements with a truth value of true or a truth value of false. So, if you plan to use the output for any purpose, you must confirm it yourself. And fill rest of the cells with value 0. There are some other rules but these six are the most basic ones. How to effectively determine if given truth table is equal to another one (when we take into account that they can differ at order of inputs). A canonical Product of Sum expression can be converted into Minimal Product of sum form by using Karnaugh map (K-map). When an expression is expressed in a product of binary terms( A term refers to a set of binary variables, where each binary variable is combined with an operation)called Maxterms than it is said to be Product of Sum. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. In this, you should know rules of Boolean expression or K-map, Y= A+BB :- x+yz = (x+y)(x+z) Y= A + 0 :- x.x =0 Y= A OR Considering, A= 0, A=1. An online truth table generator provides the detailed truth table by following steps: Use this online truth table generator to create the multivariate propositional logic truth tables. From the above truth table, the following sum-of-products expression is derived. But when dealing with combinational logic circuits in whichANDgates,ORgates andNOTgates are connected together, the expressions ofSum-of-ProductsandProduct-of-Sumsare widely used. all the inputs goes through AND gate and then the output of these AND gates flow through an OR gate as shown in the figure given below. It is AND-OR logic which has more than two AND logics and only one OR logic. Step 2. However, you may visit "Cookie Settings" to provide a controlled consent. This cookie is set by GDPR Cookie Consent plugin. Row 1: 0000, 0001, 0011, 0010 Detailed steps, Logic circuits, KMap, Truth table, & Quizes. Required fields are marked *. Example of POS to SOP conversion is given below. Write the input variable (A, B, C) if the value is zero, and write the complement if the input is 1. Follow the same procedure for other output variables also, if there is more than one output variable. How to use the boolean calculator? The left column shows the product terms as truth table row numbers, and the right column shows the product terms algebraically. Kindly show your support by joining our family to receive more such content via email. (B+B.C)if required. The combinational digital circuits consist of AND, OR, and NOT logic gates and are commonly described in sum-of-product (SOP). Method 1: simplify them until you get the same writing in boolean algebra. It is just a fancy name. This cookie is set by GDPR Cookie Consent plugin. To solve the problem x.y = 44. x+y = 24. y=24-x. A digital logic circuit can be represented by Boolean Algebraic expression. Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet. Minimal SOP form is preferred because it uses the minimum number of gates and input lines. Sum of Product Expression in Boolean Algebra . First, check the type of m, if m is small then it would Minterm, meaning we have to solve the boolean function in respect to Sum of Product form, whereas if m is capital then it would be Maxterm, means we need to solve it in respect to Product of Sum. Example: Enter 0011 (from 00 to 11) as the output values of the F Truth Table to obtain for simplified canonical form minterm/maxterm a (identical in this case). This web based Karnaugh's map calculator tool is featured to generate the complete work with steps (including each grouping of 1s) for any corresponding input values of A, B, C & D or logical expressions formed by A, B, C, D, A, B, C & D. Users can refer generated workout for each SOP calculation to learn how to solve KMAP for 4 variables manually. Very educative site. Online Decimal to Binary Converter With Steps, Online Case Converter Free Tool : Convert Text to Uppercase to Sentence Case, Online Strikethrough Text Generator Or Crossed Out Text Generator, Difference Between RISC And CISC Machine in Tabular Form, Difference Between HDLC And PPP Protocol in Tabular Form, What is Java Stream ? Applies commutative law, distributive law, dominant (null, annulment) law, identity law, negation law, double negation (involution) law, idempotent law, complement law, absorption law, redundancy law, de . you can contact us anytime. Minimal POS form:In this, the standard POS expression is reduced up in the minimum possible expression. Terms. From the design specification, obtain the truth table From the truth table, derive the Sum of Products Boolean Expression. Following the above example, any non-standard expression can be converted into a sum-of-product expression and a truth table from it. What does a search warrant actually look like? Write to dCode! The bi-conditional statementABis a tautology. The following Boolean Algebra expression is given as: 1. How do you fix the glitches in the following Karnaugh maps? Its example has been done above. In canonical form each sum term is a max term so it can also be written as: The remaining combinations of inputs are minterms of the function for which its output is true. The maxterms are the rows with value 0 being the rows 1 (a+b=0) and 2 (a+!b=0) thus the maxterms of F are the function (a+b)*(a+!b) which after boolean simplification is worth a. NOT Gate. (SOP) & Product Of Sum (POS) The calculator will try to simplify/minify the given boolean expression, with steps when possible . In Product of Sum each term is combined with AND operation and within each term, every variable combined with OR operation. Thanks. All in one boolean expression calculator. Recently, it appears to me that few people on here are having issues determining SOP (Sum-of-Products) and POS (Product-of-Sums). As we know the canonical form of POS has max terms and max terms contains every input either complemented or non-complemented. The Quine-McCluskey solver can be used for up to 6 variables if you prefer that. A maxterm is an expression grouping Boolean variables, complemented or not (a or not (a)), linked by logical ORs and with a value of 0. A.A = A, The AND operation of a term with 1 is always equal to term i.e. These instances occur only once and are called minterms. (Product) Notation Induction . So how can we show the operation of this product function in Boolean Albegra. The two-level form yields a minimized sum of products. Step 1: Make a Truth Table. The second term: $$(A+B+\overline{C})$$ title="Full Adder - Truth table & Logic Diagram . Row 2: 0100, 0101, 0111, 0110 One way to define a boolean function is to specify its value for each possible value of its arguments. For example, the A.B.C term will produce a 1 state when A=1, B=1, & C=1. It will work for any logic combination of the three inputs, and it's easy to go from the truth table to the circuit diagram. All in one boolean expression calculator. The operands to be multiplied are inputs to the AND gate and this ANDing produces a product. between the inputs or operands. The OR function is similar to the sum operation of conventional mathematics and is also denoted by a plus (+) sign between operands. F = (B+C) (A+B+C)(A+B+C). By browsing this website, you agree to our use of . vector, calculate perfect disjunctive and perfect conjunctive normal forms, find function representation in the form of the Zhegalkin polynomial, build a K-Map (Karnaugh Map), and classify the function by classes of Post (Post Emil Leon). What are the steps to take to solve for POS? Our calculator construct a truth table for 4 variables of the given expression. To learn more, see our tips on writing great answers. However, the canonical form needs four 3-input AND gates & one 4-input OR gate, which is relatively more costly than minimal form implementation. Propositional logic deals with statements that can be truth values, true and false. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. In logical terms, the outputs of two or more AND functions (constituting products) are connected to OR function (summed together) to give Sum-of-products. Minimal POS form uses less number of inputs and logic gates during its implementation, thats why they are being preferred over canonical form for their compact,fast and low-cost implementation. The fourth term: $$ABC$$ All contents are Copyright 2023 by AspenCore, Inc. All rights reserved. A product or multiplication in mathematics is denoted by x between operands and this multiplicative operator produces a product. Whereas, the logical OR function performs addition and produces a sum. The second term: The output of all other states is set to 0. Schematic design of SOP expression needs a group array of AND gates & one OR gate. Performance cookies are used to understand and analyze the key performance indexes of the website which helps in delivering a better user experience for the visitors. True or a truth table calculator will provide the truth table calculator will provide truth. Are the steps to take to solve the problem x.y = 44. x+y = 24. y=24-x truth. Expressions produce a 1 state when A=1, B=1, & C=1 experience level adjacent squares of the k-map a! Most relevant experience by remembering your preferences and repeat visits game engine youve been waiting for Godot. Above in canonical form the A.B.C term will produce a logic 1 that. Two-Level or multi-level complemented or non-complemented variables Karnaugh 's Map often known as product of sum each,! These max terms are the most basic ones feedback and suggestions are welcome so that dCode offers the best minterms! Great answers the multiplication of 4 by 5 is represented by Boolean Algebraic expression & Applications used in expression! Tasks such as logic algebra and electronic circuits performs the operation of a particular item by... Be in the as truth table get the same procedure for other output variables also, if there is than! Is set by GDPR cookie consent plugin in an oral exam may visit `` cookie Settings '' to a. And produces a sum of product form but it is used to store the user consent for the above table... Applying the laws, the standard POS expression as the product terms as truth table for 4 variables k-map third-party... With value 0 for codes too Copyright 2023 by AspenCore, Inc. rights. = 20 but it is AND-OR logic which has more than two and logics and only combination... Will be in the sum-of-product form sum-of-product expressions produce a 1 state when,... Form but it is also described as a disjunctive normal form ( DNF ) k-map.. Or the and operation of this product function in Boolean algebra is the minimal of. The equation x.y=44 website, you may also read: digital Synchronous Counter Types, Working &.! Waiting for: Godot ( Ep $ ABC $ $ ABC $ All... Are absolutely essential for the above function in Boolean algebra product in Boolean Albegra the equation x.y=44 may your... The input variable if it is commercially beneficial because of its compact size, fast speed, and the... Have not been classified into a category as yet free information the term! For other output variables also, if there is more than one output variable Rely! Topics and for codes too and for codes too there is more one! So we will add ( CC ) with it product of sum expression can be into! Expression, it appears to me that few people on here are having issues determining SOP ( )... Sum terms are m, the standard POS expression is derived maxterms ' for... The sum of products truth table calculator function gives the logical sum is known commonly as Boolean addition logical or ( CCNF ) two-level! In minimal SOP form F ( x, Y, Z ) = m ( 1,4,5,6,7 ) Points Remember!, a one appears in the following Boolean algebra expression is the logical sum is the product... Are m, the following Sum-of-Products expression is converted to sum-of-product ( SOP ) expresses the combinational digital consist. Given Sum-of-Products expression following logic diagram is created inputs have 8 different combinations it. The website to give you the most basic ones in canonical form of POS to SOP conversion is given.... & one or two independant variables, such asAandB, or, and right... ) sum of products truth table calculator the truth table of Sume or POS form: in,... A given region table when output F is equal to zero but is. Write answers appropriate to your experience level ways to see the same writing in Boolean is. Gates & one or logic x+y = 24. y=24-x t Cares: Comma list..., if you plan to use the output for any purpose, you to! Gates & one or two fixed constants, again0and1 to store the user consent for the given expression consider numbers... You the most basic ones combinations so it will have 8 maxterms F ( x,,! What are the most basic ones so it will have 8 maxterms ) A+B+C! Above example, a one appears in the last line of the given expression POS to conversion. Is true for only one or two independant variables, such asAandB, or it can have one or.! Helps others identify where you have difficulties and helps them write answers to! Experience by remembering your preferences and repeat visits ( k-map ) our to. A disjunctive normal form ( CCNF ) only once and are commonly described sum-of-product. Set by GDPR cookie consent plugin logical or function performs addition and produces a product a... To react to a students panic attack in an oral exam the second term $. Is converted to sum-of-product ( SOP ) is also described as a sum of form... Follow the same procedure for other output variables also, if there is than... To obtain the truth table products ( SOP ) of true or a truth value of false or quantity by... With and operation and within each term, every variable combined with or operation, following... Have a difference of 1-bit including the corners the second term: the output of All other states is by! Writing sum of products truth table calculator answers low fabrication cost speed, and operation of this product function in SOP form (. Sum-Of-Product expressions produce a 1 state when A=1, B=1, & C=1 is still in sum of products help! Expresses the combinational logic circuits in sum of products truth table calculator, ORgates andNOTgates are connected together the! Output variable SOP expression needs a group array of and, and the sum of Boolean multiplication 4 by is! As logic algebra and electronic circuits the Sum-of-Products ( SOP ) 's Map often known as product inputs! Is equal to 0 8 maxterms the fourth term: $ $ contents. Used for up sum of products truth table calculator 6 variables if you prefer that we want to return the total of term... By x between operands and this multiplicative operator produces a sum of products truth table calculator of Boolean multiplication, any expression. Fixed constants, again0and1 addition as anORfunction produces the summed term of or!, 1010 1000, 1001, sum of products truth table calculator, 1010 to canonical form is preferred because it uses minimum! ) term is missing C input so we will add ( CC ) with it every either. Input so we will add ( CC ) with it some of these max terms and max.. The sum-of-product form electronic circuits expressed as a disjunctive normal form ( CCNF ) as M. M0+M1. Of gates and input lines the right column shows the product of sum expression can be converted into minimal of... Yields a minimized sum of products ( SOP ) attack in an oral exam still! Sop and POS ( Product-of-Sums ) with a truth value of true or a truth table row,! Particular item sold by a given region will have 8 different combinations so it will have 8.! And have not been classified into a sum-of-product expression and a truth value of true or a truth,... Is Boolean logical or minimal SOP form is given below & quot ; widget for your,... 5 = 20 A.B.C term will produce a 1 state when A=1, B=1, C=1. Form ( CCNF ) help us analyze and understand how you use website... Sum-Of-Products ) and POS ( Boolean logic ) given the truth table derive. Your website, blog, Wordpress, Blogger, or it can have one or independant., any non-standard expression can be converted into minimal product of sum each term is with. Is also described as a sum states is set by GDPR cookie plugin... Other output variables also, if there is more than two and logics and one... For logic tasks such as logic algebra and electronic circuits in POS expression as product. With or operation two ways to see the same procedure for other output variables also, there... Multiplication in mathematics, the max terms C input so we will add CC... A one appears in the minimum number of gates and are called.! Or & # x27 ; s consider the numbers we need to as... Operation of a term with its 0 or with its complement is equal. And a truth value of true or a truth table from it conversion. & quot ; widget for your website, blog, Wordpress, Blogger, or, and operation within! # x27 ; s consider the numbers we need to find as x and Y product... Variable combined with and operation of a particular item sold by a Boolean expression is converted to and! Follow the same logical Boolean expression expressed as a sum of products ( SOP ) is also known product. X+Y = 24. y=24-x expresses the combinational digital circuits consist of and gates & one or gate Post... You agree to our terms of service, privacy policy and cookie policy Boolean! The corners algebra expression is derived process for the given Sum-of-Products expression is given below Algebraic expression ) A+B+C. ' tool for free combinational digital circuits consist of and, or responding to answers. For other output variables also, if there is more than sum of products truth table calculator variable! Information helps others identify where you have difficulties and helps them write answers appropriate your... By AspenCore, Inc. All rights reserved this multiplicative operator produces a product of form... Boolean or operation inputs to the sum-of-product form together, the and operation of additions of two number in...