Saturday, January 07, 2012
Proof Ex 3.26
Given
∀x∃y(xRy),
∀x∀y(xRy => yRx),
∀x∀y∀z(xRy ∧ yRz => xRz)
Prove that
∀x(xRx)
Proof:
Let a be arbitrary
From ∀x∃y(xRy), it follows ∃y(aRy)
Let b such that aRb
From ∀x∀y(xRy => yRx), it follows aRb => bRa
From ∀x∀y∀z(xRy ∧ yRz => xRz), it follows aRb ∧ bRa => aRa
Thus ∀x(xRx)
