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|
Operation |
Commutative |
True / False |
|
Addition |
a + (b + c) = (a + b) + c |
True |
|
Subtraction |
a − (b − c) = (a − b) − c |
False |
|
Multiplication |
a × (b × c) = (a × b) × c |
True |
|
Division |
a ÷ (b ÷ c) = (a ÷ b) ÷ c |
False |
For Integers
Let us take three integers 2, 3, 4
|
Operation |
Number |
Remark |
|
Addition
|
a + (b + c) = (a + b) + c Take a = 2, b = 3 & c = 4
L.H.S a + (b + c) = 2 + (3 + 4) = 2 + 7 = 9
∴ a + (b + c) = 9
R.H.S (a + b) + c = (2 + 3) + 4 = 5 + 4 = 9
∴ (a + b) + c = 9 |
Since a + (b + c) = (a + b) + c
∴ Addition is associative. |
|
Subtraction
|
a − (b − c) = (a − b) − c Take a = 2, b = 3 & c = 4
L.H.S a − (b − c) = 2 − (3 − 4) = 2 − (−1) = 2 + 1 = 3
∴ a − (b − c) = 3
R.H.S (a − b) − c = (2 − 3) − 4 = −1 − 4 = −(1 + 4) = −5
∴ (a − b) − c = −5 |
Since a − (b − c) ≠ (a − b) − c
∴ Subtraction is not associative. |
|
Multiplication
|
a × (b × c) = (a × b) × c Take a = 2, b = 3 & c = 4
L.H.S a × (b × c) = 2 × (3 × 4) = 2 × 12 = 24
∴ a × (b × c) = 24
R.H.S (a × b) × c = (2 × 3) × 4 = 6 × 4 = 24
∴ (a × b) × c = 24 |
Since a × (b × c) = (a × b) × c
∴ Multiplication is associative. |
|
Division
|
a ÷ (b ÷ c) = (a ÷ b) ÷ c Take a = 2, b = 3 & c = 4
L.H.S a ÷ (b ÷ c) = 2÷(3÷4) = 2÷(3/4) = 2×4/3 = 8/3
R.H.S (a ÷ b) ÷ c = (2÷3) ÷4 = (2/3)÷4 = 2/3×1/4 = 1/6
a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c |
Since a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c
∴ Division is not associative. |
