Proof of Identity Element
Definition states directly that 0 is a right identity. We prove that 0 is a left identity by induction on the natural number a.
For the base case a = 0, 0 + 0 = 0 by definition . Now we assume the induction hypothesis, that 0 + a = a. Then
- 0 + S(a)
- = S(0 + a)
- = S(a)
This completes the induction on a.
Read more about this topic: Proofs Involving The Addition Of Natural Numbers
Famous quotes containing the words proof of, proof, identity and/or element:
“Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
—Henry David Thoreau (1817–1862)
“a meek humble Man of modest sense,
Who preaching peace does practice continence;
Whose pious life’s a proof he does believe,
Mysterious truths, which no Man can conceive.”
—John Wilmot, 2d Earl Of Rochester (1647–1680)
“All that remains is the mad desire for present identity through a woman.”
—Max Frisch (1911–1991)
“The reverence for the Scriptures is an element of civilization, for thus has the history of the world been preserved, and is preserved.”
—Ralph Waldo Emerson (1803–1882)