2) A binary operation is defined on the set of real numbers R by: \[ a \triangle b = \frac{2a + \frac{1}{2}b}{2} \] Find the: (a) Inverse of -7 and 3 under the operation. (b) The identity element.

Let's analyze the operation defined on the set of real numbers R by the formula:
a △ b = 2 2 a + 2 1 ​ b ​
a) Inverse of -7 and 3 under the operation:
To find the inverse of a number under a binary operation, we need an element x such that:
a △ x = x △ a = e
where e is the identity element of the operation.
However, based on the operation formula provided, it appears a △ b might only combine two numbers, and an inverse might not exist in a traditional sense unless we fully solve or identify the operation properties.
b) The identity element:
The identity element e for a binary operation a △ b is the element that satisfies:
a △ e = a
Applying this to our operation:
a △ e = 2 2 a + 2 1 ​ e ​ = a
Solving for e :
2 2 a + 2 1 ​ e ​ = a
Multiply through by 2 to clear the fraction:
2 a + 2 1 ​ e = 2 a
Subtract 2 a from both sides:
2 1 ​ e = 0
Multiplying both sides by 2 gives:
e = 0
Thus, the identity element e is 0 for this operation.
In conclusion, while this operation does indeed have an identity element (which is 0), the structure of the given operation seems to not have a typical inverse, or may require further customization to find inverses explicitly in strange cases. The operation's setup often calls for additional specific conditions or symmetries to define 'inverses' in an algebraic structure.

Answered by danjohnbrain | 2025-07-21