Associative Law in Discrete MathematicsWe can apply the association law to the multiplication or addition of the three numbers in discrete mathematics. On the basis of this law, if there are three numbers x, y, and z, then the following relation consists between these numbers. X + (Y + Z) = (X + Y) + Z X * (Y * Z) = (X * Y) * Z With the help of above expression, we can understand that the result of addition or multiplication will not affect how we group or associate these numbers. We can apply that law to all the real numbers. There are four major arithmetic operations addition, subtraction, multiplication, and division. The association law can only be applied for two operations addition and multiplication. We cannot apply it to other arithmetic operations, i.e., division and subtraction, because when we use these operations, then the result can be changed. According to associate law, when we do the addition and multiplication operation on some integers and change the position of these integers, in this case, the sign of the integer will not be changed. Formula of Associative lawThe definition of associative law is used to determine its formula and definition. According to the definition, if we do addition or multiplication of three numbers, then these numbers will be independent of their grouping or association. In other words, the combination or grouping of three numbers at the time of addition or multiplication does not affect their result. Suppose there are three numbers, X, Y, and Z. According to this law, the following relation consists between these numbers. X + (Y + Z) = (X + Y) + Z X * (Y * Z) = (X * Y) * Z Associative law of Addition:The associative law is followed by the addition operation. Here it will not matter that how we add the numbers, but their final result will always be the same. Suppose there are three numbers, X, Y, and Z, and these numbers will contain the following relation: X + (Y + Z) = (X + Y) + Z = X + Y + Z Associative law of Multiplication:The associative law is also followed by the multiplication operation. It will not matter we clubbed the numbers here, but their final result will always be the same. If there are three numbers, X, Y, and Z, then these numbers will contain the following relation: X * (Y * Z) = (X * Y) * Z = X * Y * Z Proof of Associative lawIn the above explanation, we have discussed working on associative law. There are some examples to prove these properties. Proof of Associative Law of AdditionHere we will explain various examples to prove the associative law for addition, which are described as follows: Example 1: In this example, we have to prove that 4 + (5 + 3) = (4 + 5) + 3 Solution: To prove this, we will first solve the LHS (left hand side) part: 4 + (5 + 3) = 4 + 8 = 12 Now we will solve the RHS (right hand side) part: (4 + 5) + 3 = 9 + 3 = 12 Hence, we have proved that LHS = RHS Therefore, 4 + (5 + 3) = (4 + 5) + 3 (Proved) Example 2: In this example, we have to prove that 1 + (3 + 6) = (1 + (3)) + 6 Solution: To prove this, we will first solve the LHS (left hand side) part: 1 + (3 + 6) = 1 + 3 = 4 Now we will solve the RHS (right hand side) part: (1 + (3)) + 6 = (1  3) + 6 = 2 + 6 = 4 Hence, we have proved that LHS = RHS Therefore, 1 + (3 + 6) = (1 + (3)) + 6 (Proved) Example 3: In this example, we have to prove that 3 + (5 + 9) = (3 + (5)) + 9 Solution: To prove this, we will first solve the LHS (lefthand side) part: 3 + (5 + 9) = 3 + 4 = 1 Now we will solve the RHS (right hand side) part: (3 + (5)) + 9 = (3 + 5) + 9 = 8 + 9 = 1 Hence, we have proved that LHS = RHS Therefore, 3 + (5 + 9) = (3 + (5)) + 9 (Proved) Proof of Associative Law of MultiplicationHere we will explain various examples to prove the associative law for multiplication, which are described as follows: Example 4: In this example, we have to prove that 4 * (5 * 3) = (4 * 5) * 3 Solution: To prove this, we will first solve the LHS (left hand side) part: 4 * (5 * 3) = 4 * 15 = 60 Now we will solve the RHS (right hand side) part: (4 * 5) * 3 = 20 * 3 = 60 Hence, we have proved that LHS = RHS Therefore, 4 * (5 * 3) = (4 * 5) * 3 (Proved) Example 5: In this example, we have to prove that 2 * (3 * 6) = (2 * (3)) * 6 Solution: To prove this, we will first solve the LHS (left hand side) part: 2 * (3 * 6) = 2 * (18) = 36 Now we will solve the RHS (right hand side) part: (2 * (3)) * 6 = (6) * 6 = 36 Hence, we have proved that LHS = RHS Therefore, 2 * (3 * 6) = (2 * (3)) * 6 (Proved) Why not Division and SubtractionWe have already learned that the division and subtraction operation cannot be performed on association law. Now we will understand this with the help of some examples, which are described as follows: SubtractionSuppose there are three integers 3, 6, and 5. Now we will assume that we can apply association law on subtraction. Thus, this condition must be satisfied: 3  (6  5) = (3  6)  5 To prove this, we will first solve the LHS part: 3  (6  5) = 3  1 = 2 Now we will solve the RHS (right hand side) part: (3  6)  5 = 3  5 = 8 From LHS and RHS, we have see that 2 ≠ 8 Hence, 3  (6  5) ≠ (3  6)  5 Therefore, it is proved that the association law cannot be applied to the subtraction. DivisionSuppose there are three integers 27, 9, and 3. Now we will assume that we can apply association law on division. Thus, this condition must be satisfied: 27 / (9 / 3) = (27 / 9) / 3 To prove this, we will first solve the LHS part: 27 / (9 / 3) = 27 / 3 = 9 Now we will solve the RHS (right hand side) part: (27 / 9) / 3 = 3 / 3 = 1 From LHS and RHS, we have see that 9 ≠ 1 Hence, 27 / (9 / 3) ≠ (27 / 9) / 3 Therefore, it is proved that the association law cannot be applied to division.
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