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In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements.For example, in the conditional statement: "If P then Q", Q is necessary for P, because the truth of Q is guaranteed by the truth of P.
Logical equality is a logical operator that compares two truth values, or more generally, two formulas, such that it gives the value True if both arguments have the same truth value, and False if they are different.
In the left hand sides of the following identities, L is the L eft most set and R is the R ight most set. Whenever necessary, both L and R should be assumed to be subsets of some universe set X , so that L ∁ := X ∖ L and R ∁ := X ∖ R . {\displaystyle L^{\complement }:=X\setminus L{\text{ and }}R^{\complement }:=X\setminus R.}
The relationship x precedes y is written x ≺ y. The relation x precedes or is equal to y is written x ≼ y. The relationship x succeeds (or follows) y is written x ≻ y. The relation x succeeds or is equal to y is written x ≽ y. [citation needed]
The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The canonical map ker : X^X → Con X, relates the monoid X^X of all functions on X and Con X. ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f : X → X to its ...
The unique pair of values a, b satisfying the first two equations is (a, b) = (1, 1); since these values also satisfy the third equation, there do in fact exist a, b such that a times the original first equation plus b times the original second equation equals the original third equation; we conclude that the third equation is linearly ...
Substituting r(cos θ + i sin θ) for e ix and equating real and imaginary parts in this formula gives dr / dx = 0 and dθ / dx = 1. Thus, r is a constant, and θ is x + C for some constant C. The initial values r(0) = 1 and θ(0) = 0 come from e 0i = 1, giving r = 1 and θ = x.
It tells whether there is an odd number of 1 bits (is true if and only if an odd number of the variables are true), which is equal to the parity bit returned by a parity function. In logical circuits, a simple adder can be made with an XOR gate to add the numbers, and a series of AND, OR and NOT gates to create the carry output.