Solved Examples on Commutative Law

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].

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.