(R^-1)^-1 = R Theorem for Relations Proof [ILIEKMATHPHYSICS]

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Proof of R = (R^-1)^-1 for relations (ILIEKMATHPHYSICS)

Proof that (S ∘ R)^-1 = R^-1 ∘ S^-1 for relations (ILIEKMATHPHYSICS)

Proof that Dom(R) = Ran(R^-1) and Dom(R^-1) = Ran(R) for relations (ILIEKMATHPHYSICS)

R is symmetric iff R = R^-1 Proof [ILIEKMATHPHYSICS]

(x,y) ∈ R iff (y,x) ∈ R^-1 (Proof) [ILIEKMATHPHYSICS]

Prove that 0 ﹤ 1/2 ﹤ 1 (ILIEKMATHPHYSICS)

R is transitive iff R ∘ R ⊆ R Proof [ILIEKMATHPHYSICS]

Every real number has exactly one negative (Proof) [ILIEKMATHPHYSICS]