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Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations3 Marks
Question
Let '*' be a binary operation on N defined by a * b = 1.c.m. (a, b) for all $\text{a, b}\in\text{N}.$
Check the commutativity and associativity of '*' on N.
✓

Answer

Commutativity: Let $\text{a, b}\in\text{N}$

a * b = 1.c.m. a, b

= 1.c.m. b, a

= b * a

Therefore,

$\text{a}\ ^*\ \text{b}=\text{b}\ ^*\ \text{a}\ \forall\ \text{a, b}\in\text{N}$

Thus, * is Commutative on N.

Associativity: Let $\text{a, b, c}\in\text{N}$

a * b * c = a * 1.c.m. b, c

= 1.c.m. a, b, c

a * b * c = 1.c.m. a, b * c

= 1.c.m. a, b, c

Therefore,

$\text{a}\ ^*\ \text{b}\ ^*\ \text{c}=\text{a}\ ^*\ \text{b}\ ^*\ \text{c}\ \forall\ \text{a, b, c}\in\text{N}$

Thus, * is associative on N.

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