Distributive Laws
For sentences p,q,r:
p ∧ ( q ∨ r ) ≡ ( p ∧ q ) ∨ ( p ∧ r )
p ∨ ( q ∧ r ) ≡ ( p ∨ q ) ∧ ( p ∨ r )
Or, using Tem42's Everything Logic Symbols if the above hasn't rendered correctly on your browser:
p * ( q ^ r ) == ( p * q ) ^ ( p * r )
p ^ ( q * r ) == ( p ^ q ) * ( p ^ r )
The first law can be shown to be true by comparing columns 5 and 8 of the following truth table:
p q r | q∨r p∧(q∨r) p∧q p∧r (p∧q)∨(p ∧ r)
T T T | T T T T T
T T F | T T T F T
T F T | T T F T T
T F F | F F F F F
F T T | T F F F F
F T F | T F F F F
F F T | T F F F F
F F F | F F F F F
To prove the second law holds, consider the negation of each side, using DeMorgan's Laws:
negation of LHS= ¬(p∨(q∧r)) ≡ (¬p)∧(¬(q∧r)) ≡ (¬p)∧((¬q)∨(¬r))
negation of RHS= ¬((p∨q)∧(p∨r)) ≡ (¬(p∨q))∨(¬(p∨r)) ≡ ((¬p)∧(¬q))∨((¬p)∧(¬r))
From the first distributive law, the right-hand sides of the above two expressions are equivalent. Thus the negations of the L- and RHS of the second distributive law are equal: thus they are equal (negate again and the negations cancel, leaving the L- and RHS) and the second law must hold.
The distributive laws also hold for sets, with union in place of logical or, and intersection in place of logical and:
p ∩ ( q ∪ r ) ≡ ( p ∩ q ) ∪ ( p ∩ r )
p ∪ ( q ∩ r ) ≡ ( p ∪ q ) ∩ ( p ∪ r )