View source for Equivalence (logical)

''of two propositions (formulas) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036040/e0360401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036040/e0360402.png" />''

Two propositions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036040/e0360403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036040/e0360404.png" /> are (logically) equivalent if for each admissible choice of values of the parameters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036040/e0360405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036040/e0360406.png" /> either both are true or both are false. For example, equivalence of equations, inequalities and systems of them means the coincidence of their solution sets. Equivalence of formulas of [[Propositional calculus|propositional calculus]] is the coincidence of the Boolean functions (cf. [[Boolean function|Boolean function]]) that they define.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Novikov,   "Elements of mathematical logic" , Oliver &amp; Boyd and Acad. Press  (1964)  (Translated from Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.C. Kleene,   "Mathematical logic" , Wiley  (1967)</TD></TR></table>