what is communicative property associative property distributive property and closure property
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Commutative Property
The word, Commutative, originated from the French word “commute or commuter” means to switch or move around combined with the suffix “-ative” means tend to. Therefore, literal meaning of word is tending to switch or move around.
The commutative property states that if we swipe the positions of the numbers result will remain the same.
Associative Property
According to the associative law, regardless of how the numbers are grouped, you can add or multiply them. In other words, the placement of parentheses does not matter when it comes to adding or multiplying.
Distributive Property of Multiplication
The distributive property of Multiplication states that multiplying a sum by a number is the same as multiplying each addend by the value and adding the products then.
According to the Distributive Property, if a, b, c are real numbers:
a x (b + c) = (a x b) + (a x c)
Non-Commutative Property
Some operations are non-commutative. By non-commutative we mean the switching of order will give different results. The mathematical operations, subtraction and division are the two non-commutative operations. Unlike addition, in subtraction switching of orders of terms results in different answers.
Example, 4 − 3 = 1 but 3 − 4 = −1 which are two different integers.
Also, division does not follow the commutative property. That is,
6 ÷ 2 = 3
2 ÷ 6 = 13
⇒ 6 ÷ 2 ≠ 2 ÷ 6
Important Note: Commutative property works for addition and multiplication only but not for subtraction and division.
Examples
Example 1: Which of the following follows commutative law?
3 × 12
4 + 20
36 ÷ 6
36 − 6
−3 × 4
Solution: Options 1, 2 and 5 follows the commutative law
Explanation:
3 × 12 = 36 and
12 x 3 = 36
=> 3 x 12 = 12 x 3 (commutative)
4 + 20 = 24 and
20 + 4 = 24
=> 4 + 20 = 20 + 4 (commutative)
36 ÷ 6 = 6 and
6 ÷ 36 = 0.17
=> 36 ÷ 6 ≠ 6 ÷ 36 (non commutative)
36 − 6 = 30 and
6 – 36 = – 30
=> 36 – 6 ≠ 6 – 36 (non commutative)
−3 × 4 = -12 and
4 x -3 = -12
=> −3 × 4 = 4 x -3 (commutative)
Closure property
A set is closed under an operation if performance of that operation on members of the set always produces a member of that set. For example, the positive integers are closed under addition, but not under subtraction:
1
−
2
1-2 is not a positive integer even though both 1 and 2 are positive integers. Another example is the set containing only zero, which is closed under addition, subtraction and multiplication (because
0
+
0
=
0
0+0=0,
0
−
0
=
0
0-0=0, and
0
×
0
=
0
0\times {0}=0).
Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operations individually.
The word, Commutative, originated from the French word “commute or commuter” means to switch or move around combined with the suffix “-ative” means tend to. Therefore, literal meaning of word is tending to switch or move around.
The commutative property states that if we swipe the positions of the numbers result will remain the same.
Associative Property
According to the associative law, regardless of how the numbers are grouped, you can add or multiply them. In other words, the placement of parentheses does not matter when it comes to adding or multiplying.
Distributive Property of Multiplication
The distributive property of Multiplication states that multiplying a sum by a number is the same as multiplying each addend by the value and adding the products then.
According to the Distributive Property, if a, b, c are real numbers:
a x (b + c) = (a x b) + (a x c)
Non-Commutative Property
Some operations are non-commutative. By non-commutative we mean the switching of order will give different results. The mathematical operations, subtraction and division are the two non-commutative operations. Unlike addition, in subtraction switching of orders of terms results in different answers.
Example, 4 − 3 = 1 but 3 − 4 = −1 which are two different integers.
Also, division does not follow the commutative property. That is,
6 ÷ 2 = 3
2 ÷ 6 = 13
⇒ 6 ÷ 2 ≠ 2 ÷ 6
Important Note: Commutative property works for addition and multiplication only but not for subtraction and division.
Examples
Example 1: Which of the following follows commutative law?
3 × 12
4 + 20
36 ÷ 6
36 − 6
−3 × 4
Solution: Options 1, 2 and 5 follows the commutative law
Explanation:
3 × 12 = 36 and
12 x 3 = 36
=> 3 x 12 = 12 x 3 (commutative)
4 + 20 = 24 and
20 + 4 = 24
=> 4 + 20 = 20 + 4 (commutative)
36 ÷ 6 = 6 and
6 ÷ 36 = 0.17
=> 36 ÷ 6 ≠ 6 ÷ 36 (non commutative)
36 − 6 = 30 and
6 – 36 = – 30
=> 36 – 6 ≠ 6 – 36 (non commutative)
−3 × 4 = -12 and
4 x -3 = -12
=> −3 × 4 = 4 x -3 (commutative)
Closure property
A set is closed under an operation if performance of that operation on members of the set always produces a member of that set. For example, the positive integers are closed under addition, but not under subtraction:
1
−
2
1-2 is not a positive integer even though both 1 and 2 are positive integers. Another example is the set containing only zero, which is closed under addition, subtraction and multiplication (because
0
+
0
=
0
0+0=0,
0
−
0
=
0
0-0=0, and
0
×
0
=
0
0\times {0}=0).
Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operations individually.
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