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Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.
The rule states that P implies Q is logically equivalent to not-or and that either form can replace the other in logical proofs. In other words, if P {\displaystyle P} is true, then Q {\displaystyle Q} must also be true, while if Q {\displaystyle Q} is not true, then P {\displaystyle P} cannot be true either; additionally, when P {\displaystyle ...
Logical equivalence is different from material equivalence. Formulas p {\displaystyle p} and q {\displaystyle q} are logically equivalent if and only if the statement of their material equivalence ( p ↔ q {\displaystyle p\leftrightarrow q} ) is a tautology.
In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs.It is also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions-or formulae-as-types interpretation.
Venn diagram of (true part in red) In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements and to form the statement "if and only if" (often abbreviated as "iff " [1]), where is known as the antecedent, and the consequent.
Proof (1) If L {\displaystyle L} is regular, construct a minimal DFA to accept it. Clearly, if x , y {\displaystyle x,y} end up in the same state after running through the DFA, then x ∼ L y {\displaystyle x\sim _{L}y} , thus the number of equivalence classes of ∼ L {\displaystyle \sim _{L}} is at most the number of DFA states, which must be ...
5.5 Proof of basic properties. 6 Similar relations. ... The equivalence relation of equality is a special case, as, if restricted to a given set , it is the ...