$ \begin{align*} x \cdot y &= y \cdot x \\ x + y &= y + x \\ x \cdot (y\cdot z) &= (x\cdot y)\cdot z\\ x+(y+z) &=(x+y)+z\\ x\cdot (y+z) &= (x\cdot y)+(x\cdot z)\\ x+(y\cdot z) &= (x+y)\cdot(x+z)\\ x\cdot x &= x\\ x+x &= x\\ x\cdot(x+y) &= x\\ x+(x\cdot y) &= x\\ x\cdot x\prime &= 0\\ x+x\prime &= 1\\ (x\prime)\prime &= x\\ (x\cdot y)\prime &= x\prime + y\prime \\ (x+y)\prime &= x\prime \cdot y\prime \\ \end{align*} $

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