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Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations4 Marks
Question
Consider the binary operation * and o defined by the following tables on set S = {a, b, c, d}.
*
a
b
c
d
a
a
b
c
d
b
b
a
d
c
c
c
d
a
b
d
d
c
b
a
✓

Answer

Commutativity:

The table is symmetrical about the leading element. It means * is commutative on S.

Associativity:

a * (b * c) = a * d

= d

(a * b) * c = b * c

=d

Therefore,

a * (b * c) = (a * b) * c $\forall\text{ a, b, c}\in\text{S}$

So, * is Associative on S.

Finding identity element:

We observe that the first row of the composition table coincides with the top-most row and the first column coincides with the left-most column.

These two intersect at a.

⇒ x * a = a * x = x, $\forall\text{ x}\in\text{S}$

So, a is the identity element:

a * a = a

⇒ a-1 = a

b * b = a

⇒ b-1 = b

c * c = a

⇒ c-1 = c

d * d = a

⇒ d-1 = d

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