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In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of and is sometimes expressed as , ::, , or , depending on the notation being used.
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.
To avoid confusion with the usual equality and membership, these are denoted by ‖ x = y ‖ and ‖ x ∈ y ‖ for x and y in V B. They are defined as follows: ‖ x ∈ y ‖ is defined to be Σ t ∈ Dom(y) ‖ x = t ‖ ∧ y(t) ("x is in y if it is equal to something in y"). ‖ x = y ‖ is defined to be ‖ x ⊆ y ‖∧‖ y ⊆ x ...
The number e (e = 2.71828...), also known as Euler's number, which occurs widely in mathematical analysis The number i , the imaginary unit such that i 2 = − 1 {\displaystyle i^{2}=-1} The equation is often given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.
The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q).
If E is a logical predicate, means that there exists at least one value of x for which E is true. 2. Often used in plain text as an abbreviation of "there exists". ∃! Denotes uniqueness quantification, that is, ! means "there exists exactly one x such that P (is true)".
A similar problem, involving equating like terms rather than coefficients of like terms, arises if we wish to de-nest the nested radicals + to obtain an equivalent expression not involving a square root of an expression itself involving a square root, we can postulate the existence of rational parameters d, e such that
E is a high-performance theorem prover for full first-order logic with equality. [1] It is based on the equational superposition calculus and uses a purely equational paradigm. It has been integrated into other theorem provers and it has been among the best-placed systems in several theorem proving competitions.