Minimal Transitive Closure Any binary relation over any set (finite or infinite) must has a transitive closure. Moreover, every binary relation must has a minimal transitive closure. Who proved this well-known result in abstract algebra first? Many abstract algebra texts include this.

TRANSITIVE. (Of a binary relation) Bertrand Russell wrote in "On the Notion of Order," _Mind_ , **10** , (1901), p. 32. "When ARB and BRC imply ARC, I call R _transitive_ .... This term was used in this sense by De Morgan ... It is now generally adopted." De Morgan wrote in "On the Symbols of Logic, the Theory of the Syllogism, and in particular of the Copula," _Transactions of the Cambridge Philosophical Society_ , **9** , (1850) p. 104: "The first is what I shall call _transitiveness_ , symbolized in X—Y—Z = X—Z; meaning that if X stand in the relation denoted by — to Y, and Y to Z, X therefore stands in that relation to Z." (OED)

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