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Question
Let * be an operation defined as *: R × R ⟶ R, a * b = 2a + b, a, b ∈ R. Check if * is a binary operation. If yes, find if it is associative too.
Solution
Let a, b ∈ R. Then,
a + b ∈ R ......(Addition is a binary operation on R)
⇒ a + (a+ b) ∈ R .........(Addition is a binary operation on R)
⇒ 2a + b ∈ R
Thus, a*b ∈ R for all a, b ∈ R.
Hence, * is a binary operation on R.
Let a, b,c ∈ R
(a * b) * c = (2a + b) * c = 2 (2a+ b) + c = 4a + 2b + c
a* (b * c) = a* (2b + c) = 2a + 2b + c
Since (a * b) * c ≠ a * (b * c),
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