Boolean Algebra

Obvious axioms

• $0\cdot 0=0$
• $1\cdot 1=1$
• $0\cdot 1=1\cdot 0=0$
• If $x=1$, then $\bar{x}=0$ Extrapolated axioms:
• $x\cdot 0=0$
• $x\cdot 1=x$
• $x\cdot x=x$
• $x\cdot \bar{x}=0$
• $\overline{\stackrel{ˉ}{x}}=x$
• $x+1=1$
• $x+0=x$
• $x+x=x$
• $x+\bar{x}=1$ Laws
• Commutative Law
  • $x\cdot y=y\cdot x$
  • $x+y=y+x$
• Associative
  • $x\cdot (y\cdot z)=(x\cdot y)\cdot z$
  • $x+(y+z)=(x+y)+z$
• Distributive
  • $x\cdot (y+z)=x\cdot y+x\cdot z$
  • $x+(y\cdot z)=(x+y)\cdot (x+z)$
    • In “conventional algebra” (like literally $+$ and $\times$ from elementary) this law would make no sense.
• Simplification Law
  • $x+(\bar{x}\cdot y)=x+y$
  • $x\cdot (\bar{x}+y)=x\cdot y$
• Consensus law
  • 
  • todo It’s kinda weeirrdd
• Absorption Law
  • $x\cdot (x+y)=x$
  • $x+(x\cdot y)=x$
• De Morgan’s Law
  • You know this one already
  • However there is also “bubble pushing”
  • 

$B\cdot C+\bar{B}\cdot \bar{A}+\bar{B}\cdot \bar{C}$