give one example where the associative property of subtraction does not hold good
Answers
Answer:
What are Integers?
IntegersAn integer is any number which is included in a set of combination of all the numbers types. i.e. if we construct a set of all natural numbers, whole numbers and negative numbers then such a set is called an Integer set. Integer is represented as
(……-3, -2, -1, 0, 1, 2, 3, ……)
So, every natural number is an integer, every whole number is an integer, and every negative number is an integer.
Properties of Integers
The three properties of integers are:
Closure Property
Commutativity Property
Associative Property
Let us now study these properties in detail.
Closure Property
The System of Integers in Addition
It states that addition of two Integers always results in an Integer. For example, 7 + 4 = 11, the result we get is an integer. Therefore, the system is closed under addition.
The System of Integers under Subtraction
It states that subtraction of two Integers always results in an Integer. For example, 7 – 4 = 3, the result we get is an integer. Also, 2 – 4 = -2. The result is also an integer. Therefore, the system is closed under subtraction.
The System of Integers under Multiplication
It states that multiplication of two integers always results in an integer. For example, 7 × 4 = 28, the result we get is an integer. Therefore, the system is closed under multiplication.
The System of Integers under Division
It states that division of two integers does not always result in an integer. For example, 7 ÷ 4 = 74, the result we get is not an integer. But, 8 ÷ 4 = 2, the result we get is an integer. Therefore, a system is not closed under division.Commutative Property
It is a property that associates with binary operations or functions like addition, multiplication. Take any two numbers a and b and subtract them. That is a – b, say 5 – (-3). Now subtract a from b. That is b – a or -3 – 5. Are they same? No, they are not equal. So, the commutative property does not hold for subtraction. Similarly, it does not hold for division too.
Again take any two numbers a and b and add them. That is a + b. Now add b and a which comes to be b+ a. Aren’t the same? Yes, they are equal because of commutative property which says that we can swap the numbers and still we get the same answer.
Associative Property
Associative property of integers states that for any three elements(numbers) a, b and c
1) For Addition a + ( b + c ) = ( a + b ) + c
For example, if we take 2 , 5 , 11
2 + ( 5 + 11 ) = 18 and ( 2 + 5 ) + 11 = 18
2) For Multiplication a × ( b × c ) = ( a × b ) × c
For example, 2 × ( 5 × 11 ) = 110 and ( 2 × 5 ) × 11 = 110
Hence associative property is true for addition and multiplication.
3) For Subtraction
Associative property does not hold for subtraction a – ( b – c ) != ( a – b ) – c
For example, if we take 4, 6, 12
4 – ( 6 – 12 ) and ( 4 – 6 ) – 12
= 4 + 6 = 10 and -2 -=12 = – 14
Therefore associative property is not true for subtraction.
4) For DivisionAssociative property does not hold for division a ÷ ( b ÷ c ) != ( a ÷ b ) ÷ c
For example, again if we take 4, 6, 12
4 ÷( 6 ÷12 ) and ( 4 ÷ 6 ) ÷ 12
= 4 ÷ 612 and 46 ÷ 12
we get, = 4 × 2 = 8 and 13×6 = 118
Therefore associative property is not true for division.
Multiplicative Identity for Integers
The multiplicative identity of any integer a is a number b which when multiplied with a, leaves it unchanged, i.e. b is called as the multiplicative identity of any integer a if a× b = a. Now, when we multiply 1 with any of the integers a we get a × 1 = a = 1 × a So, 1 is the multiplicative identity for integers.Additive Identity for Integers
The additive identity of any integer a is a number b which when added with a, leaves it unchanged, i.e. b is called as the additive identity of any integer a if a + b = a. Now, when we add 0 with any of the integers a we get a + 0 = a = 0 + a So, 0 is the additive identity for integers.
Solved Examples for YouQuestion: As per the commutative property, complete the equation for integers for the given operation: 10 × 5 =
A.500 B.5 × 10 C.– 50 D.None of the above Solution: B. 10 × 5 = 5 × 10 = 50. This is a commutative property of multiplication.
Question: Positive and negative integers together are closed under …………
A. addition and subtraction
B. addition and multiplication
C. multiplication and division
D. multiplication Solution: D. The set of positive and negative integers does not include 0. Addition of two integers may result in 0. Also, subtraction of two integers may result in 0. We also know integers are not closed under division.
The associative property is helpful while adding or multiplying multiple numbers. By grouping, we can create smaller components to solve. It makes the calculations of addition or multiplication of multiple numbers easier and faster. Here, adding 17 and 3 gives 20.