Question: Transforming between two encodings for Boolean vectors. A Boolean \(n\) -vector is one for which all entries are either 0 or $1 .$ Such vectors are used to encode whether each of \(n\) conditions holds, with \(a_{i}=1\) meaning that condition \(i\) holds. Another common encoding of the same information uses the two values \(-1\) and \(+1\) for the entries. For example the Boolean vector \((0,1,1,0)\) would be written using this alternative encoding as \((-1,+1,+1,-1)\). Suppose that \(x\) is a Boolean vector with entries that are 0 or 1 , and \(y\) is a vector encoding the same information using the values \(-1\) and \(+1\). Express \(y\) in terms of \(x\) using vector notation. Also, express \(x\) in terms of \(y\) using vector notation.

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