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A property of a pair of binary operations. We say that the operation @ distributes on the left over the operation \$ if (x@y)\$z = (x\$z) @ (y\$z), and that it distributes on the right if x\$(y@z) = (x\$y)@(x\$z) (if \$ is commutative, each type implies the other, and we simply say the @ distributes over \$).

For example, addition distributes over multiplication, but multiplication doesn't distribute over addition.

This simple algebraic process is also called expanding brackets, the opposite of factorisation, a process of simplifying equations.

### 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 )

Dis*trib"u*tive (?), a. [Cf. F. distributif.]

1.

Tending to distribute; serving to divide and assign in portions; dealing to each his proper share.

"Distributive justice."

Swift.

2. Logic

Assigning the species of a general term.

3. Gram.

Expressing separation; denoting a taking singly, not collectively; as, a distributive adjective or pronoun, such as each, either, every; a distributive numeral, as (Latin) bini (two by two).

Distributive operation Math., any operation which either consists of two or more parts, or works upon two or more things, and which is such that the result of the total operation is the same as the aggregated result of the two or more partial operations. Ordinary multiplication is distributive, since a × (b + c) = ab + ac, and (a + b) × c = ac + bc. -- Distributive proportion. Math. See Fellowship.

Dis*trib"u*tive, n. Gram.

A distributive adjective or pronoun; also, a distributive numeral.

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