Commutative Law Worksheet

Commutative Law in mathematics states that the order of elements does not affect the result of certain operations. This article explains the Commutative Law in mathematics for addition, multiplication, percentages, sets, and related concepts with proofs and examples.

Commutative Law states that for certain mathematical operations like addition and multiplication, changing the order of the numbers being operated on does not change the result.

Stated otherwise, A + B equals B + A, and A × B equals B × A.

Commutative Law of Addition

According to the Commutative Law of Addition, swapping the order of a and b, i.e., adding b first and then a, yields the same result c.

Mathematically, this is represented as a + b = b + a. This holds true because addition is essentially combining quantities, and the order in which we combine them doesn’t affect the total.

Commutative Law of Multiplication

Similar to addition, the Commutative Law of Multiplication states that changing the order of the numbers being multiplied doesn’t change the result.

Mathematically, this is represented as a × b = b × a. This holds true because multiplication represents repeated addition, and the order of the factors doesn’t affect the total product.

Commutative Law of Sets

The Commutative Law of Sets states that for any two sets A and B, the union (A ∪ B) and the intersection (A ∩ B) operations are commutative. In other words, the order in which sets are combined (union) or intersected does not affect the result. Mathematically, this is represented as:

• Union: A ∪ B = B ∪ A
• Intersection: A ∩ B = B ∩ A

Addition

Commutative Law: Changing the order of operands does not affect the result.
Example: 3 + 5 = 5 + 3
Solution:
3 + 5 = 8
5 + 3 = 8
Therefore, 3 + 5 = 5 + 3 = 8.

Multiplication

Commutative Law: Changing the order of factors does not affect the result.
Example: 2 x 4 = 4 x 2
Solution:
2 x 4 = 8
4 x 2 = 8
Therefore, 2 x 4 = 4 x 2 = 8.

Matrix Addition

Commutative Law: Changing the order of matrices does not affect the result.
Example:
A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]
A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]]
B + A = [[5+1, 6+2], [7+3, 8+4]] = [[6, 8], [10, 12]]
Therefore, A + B = B + A = [[6, 8], [10, 12]].

Vector Addition

Commutative Law: Changing the order of vectors does not affect the result.
Example:
v = [1, 2, 3], w = [4, 5, 6]
v + w = [1+4, 2+5, 3+6] = [5, 7, 9]
w + v = [4+1, 5+2, 6+3] = [5, 7, 9]
Therefore, v + w = w + v = [5, 7, 9].

Intersection of Sets

Commutative Law: The order of sets does not affect the result of intersection.
Example: A ∩ B = B ∩ A
If A = {1, 2, 3} and B = {3, 4, 5},
A ∩ B = {3}
B ∩ A = {3}
Therefore, A ∩ B = B ∩ A = {3}.

Union of Sets

Commutative Law: The order of sets does not affect the result of union.
Example: A ∪ B = B ∪ A
If A = {1, 2, 3} and B = {3, 4, 5},
A ∪ B = {1, 2, 3, 4, 5}
B ∪ A = {1, 2, 3, 4, 5}
Therefore, A ∪ B = B ∪ A = {1, 2, 3, 4, 5}.

Function Composition

Commutative Law: Changing the order of function composition does not affect the result.
Example: (f ∘ g)(x) = (g ∘ f)(x)
Solution:
Let f(x) = x² and g(x) = 2x
(f ∘ g)(x) = f(g(x)) = f(2x) = (2x)² = 4x²
(g ∘ f)(x) = g(f(x)) = g(x² ) = 2x²
Therefore, (f ∘ g)(x) = (g ∘ f)(x) = 4x² = 2x² .

Function Addition

Commutative Law: Changing the order of function addition does not affect the result.
Example: (f + g)(x) = (g + f)(x)
Solution:
Let f(x) = 2x and g(x) = 3x
(f + g)(x) = f(x) + g(x) = 2x + 3x = 5x
(g + f)(x) = g(x) + f(x) = 3x + 2x = 5x
Therefore, (f + g)(x) = (g + f)(x) = 5x.

Scalar Multiplication

Commutative Law: Changing the order of scalar multiplication does not affect the result.
Example: 2 x (3 x v) = (2 x 3) x v
Solution:
Let v = [1, 2, 3]
2 x (3 x v) = 2 x [3, 6, 9] = [6, 12, 18]
(2 x 3) x v = 6 x [1, 2, 3] = [6, 12, 18]
Therefore, 2 x (3 x v) = (2 x3) x v = [6, 12, 18].

Union of Intervals

Commutative Law: Changing the order of interval union does not affect the result.
Example: [1, 3] ∪ [4, 6] = [4, 6] ∪ [1, 3]
Solution:
[1, 3] ∪ [4, 6] = [1, 3] ∪ [4, 6] = [1, 6]
[4, 6] ∪ [1, 3] = [4, 6] ∪ [1, 3] = [1, 6]
Therefore, [1, 3] ∪ [4, 6] = [4, 6] ∪ [1, 3] = [1, 6].

Question 1: Check the commutative laws of multiplication for the numbers 6 and 7 and addition for the numbers 5 and 9.

Question 2: Check that the addition commutative property holds for the integers -3 and 4.

Question 3: Check that the multiplication commutative property holds for the integers -2 and 10.

Question 4: Check the 3/4 and 1/4 fractions’ commutative rule of addition.

Question 5:Check the 2/5 and 5/2 fractions’ commutative rule of multiplication.

Question 6: Check that addition for the integers 0 and 12 follows the commutative rule.

Question 7: Check that 1 and 15 follow the commutative rule of multiplication.

Question 8: Check that the addition commutative property holds for the integers -7 and 7.

Question 9:Check the multiplication commutative rule for the integers -4 and -6.

Read More,

• Distributive Property
• Associative Law of Multiplication
• Associative Law
• Associative Law Worksheet

What is the Commutative Law of Sets?

The order that sets are intersected or joined (union) has no effect on the outcome, according to the Commutative Law of Sets.

What are the operations to which the Commutative Law applies?

The Commutative Law applies to addition and multiplication. For addition, it states that a + b = b + a, and for multiplication, it states that a × b = b × a

Does the Commutative Law apply to subtraction and division?

No, the Commutative Law does not apply to subtraction and division.

How does the Commutative Law help simplify calculations?

The Commutative Law allows us to rearrange terms in an expression without changing its value. This property can make calculations more straightforward and can be particularly useful when dealing with algebraic expressions.