Some authors do not include the multiplicative identity element in their requirements for a ring ( " see ring " ), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.
32.
While some of the familiar rules of arithmetic algebra continue to hold in the algebra of propositions ( e . g . the commutative and associative laws for AND and OR ), some do not ( e . g . the distributive laws for AND, OR and NOT ).
33.
Two terms with the same indeterminates raised to the same powers are called " similar terms " or " like terms ", and they can be combined, using the distributive law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined.
34.
But this is nothing more than the geometric version of the ( left ) distributive law, a ( b + c + d ) = ab + ac + ad; and in Books V and VII of the " Elements " the commutative and associative laws for multiplication are demonstrated.
35.
Some students simply are not " convinced " by algebraic descriptions or an explanation of the distributive law, " even though these are the " modalities of choice " for a trained mathematician ", even though these are ( ostensibly ) the principles that you are intending to convey to your students.
36.
This basis makes for an easy approach to proof, called calculation, that proceeds by simplifying expressions to 0 or 1, by invoking axioms ( 2 ) & ndash; ( 4 ), and the elementary identities AA = A, \ overline { \ overline { A } } = A, 1 + A = 1, and the distributive law.
37.
"" The distributive law in mathematics is the law relating the operations of multiplication and addition, stated symbolically, a * ( b + c ) = a * b + a * c; that is, the monomial factor a is distributed, or separately applied, to each term of the binomial factor b + c, resulting in the product a * b + a * c " "-Britannica
38.
Here we try to explain the complexity of solving the MPF problem in terms of the number of mathematical operations required for the calculation . i . e . We compare the number of operations required when calculated using the normal method ( Here by normal method we mean by methods that do not use message passing or junction trees in short methods that do not use the concepts of GDL ) and the number of operations using the generalized distributive law.
39.
If the lattice is defined in terms of the order instead, i . e . ( A, d " ) is a bounded partial order with a least upper bound and greatest lower bound for every pair of elements, and the meet and join operations so defined satisfy the distributive law, then the complementation can also be defined as an involutive anti-automorphism, that is, a structure " A " = ( A, d ", ?) such that:
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