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In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality (+) = + is always true in elementary algebra. For example, in elementary arithmetic , one has 2 ⋅ ( 1 + 3 ) = ( 2 ⋅ 1 ) + ( 2 ⋅ 3 ) . {\displaystyle 2\cdot (1+3)=(2\cdot 1)+(2\cdot 3).}
2.1 Distributive properties. 2.2 First ... where i, j, and k are the unit vectors for the x-, y-, ... formula to Riemannian manifolds is a defining property of a ...
The map φ defined by φ(y) = x ∨ y is a lattice homomorphism from L to the upper closure ↑x = { y ∈ L: x ≤ y}; The binary relation Θ x on L defined by y Θ x z if x ∨ y = x ∨ z is a congruence relation, that is, an equivalence relation compatible with ∧ and ∨. [3] In an arbitrary lattice, if x 1 and x 2 are distributive ...
A lattice (L,∨,∧) is distributive if the following additional identity holds for all x, y, and z in L: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty finite joins. It is a basic fact of lattice theory that the above condition is equivalent to its ...
The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra. The term appears in William Betz's 1929 text Algebra for Today, where he states: [2]
[2] The cross product is anticommutative (that is, a × b = − b × a) and is distributive over addition, that is, a × (b + c) = a × b + a × c. [1] The space together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
The following proposition says that for any set , the power set of , ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra.
Conversely, if this "distributive property" holds for all non-negative real numbers, and , then the set is convex. [6] An example of a non-convex set such that +. The figure to the right shows an example of a non-convex set for which +.