Boolean Algebra

Boolean Algebra is useful for describing logic circuits

Boolean §

$x,y,z\in \mathbb{B}$

Postulate = Axioms

• Postulate 1: Closure with respect to (wrt) ’+’ and ’$\cdot$’
  • $x+y\in \mathbb{B}$ , $x\cdot y\in \mathbb{B}$
• Postulate 2: An identity element wrt ’+’, denoted by 0, and an identity element wrt ’$\cdot$’ denoted by 1
  • $x+0=x$ , $x\cdot 1=x$
• Postulate 3: Commutative wrt ’+’ and ’$\cdot$’
  • $x+y=y+x$ , $x\cdot y=y\cdot x$
• Postulate 4: Distributive over ’+’ and ’$\cdot$’
  • $x\cdot (y+z)=x\cdot y+x\cdot z,x+y\cdot z=(x+y)\cdot (x+z)$
• Postulate 5: For each element $a\in \mathbb{B}$ there exists an element $a^{\prime }\in \mathbb{B}$ such that
  • $a+a^{\prime }=1$ and $a\cdot a^{\prime }=0$
• Postulate 6: There exists at least 2 elements $a,b\in \mathbb{B}$ such that $a\ne b$