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I'm learning about logic circuits and I've come across full adder. In the book they derived its two carry out expressions -
Cout = x&&y || x&&z || y&&z and Cout = x&&y || (x'&&y || x&&y')&&z
I've tried to get the second equation from the first one but couldn't. Any one knows how to do that?
From your question it is unclear what $x$, $y$ and $z$ are. However, the two formulas are equivalent. Let me use products to denote a logical "AND" and additions to denote a logical "OR" so that your first formula becomes $xy + xz + yz$
If you gather $z$ in the last two terms you get: $xy + (x+y)z$.
Notice how the second term $(x+y)z$ is true iff $z$ is true, and at least one of $x$ and $y$ is true. However, if both $x$ and $y$ are true, the whole formula is true regardless of the truth value of the second term (since the first term is $xy$). You can then replace $(x+y)$ with $(x \oplus x)$ (where "$\oplus$" denotes the exclusive OR) to obtain: $xy + (x \oplus y)z$ or, equivalently, $xy + (x'y + xy')z$.
