An operation * is defined on the set, R, of real numbers by \(p * q = p + q + 2pq\). If the identity element is 0, find the value of p for which the operation has no inverse.
Given the formula for p * q as: \(p + q + 2pq\) and its identity element is 0, such that if, say, t is the inverse of p, then
\(p * t = 0\), then \(p + t + 2pt = 0 \therefore p + (1 + 2p)t = 0\)
\(t = \frac{-1}{1 + 2p}\) is the formula for the inverse of p and is undefined on R when
\(1 + 2p) = 0\) i.e when \(2p = -1; p = \frac{-1}{2}\).
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