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)
“Talk shows are proof that conversation is dead.”
—Mason Cooley (b. 1927)
“Though your views are in straight antagonism to theirs, assume an identity of sentiment, assume that you are saying precisely that which all think, and in the flow of wit and love roll out your paradoxes in solid column, with not the infirmity of a doubt.”
—Ralph Waldo Emerson (1803–1882)
“Lisa Fremont: Surprise is the most important element of attack. And besides, you’re not up on your private eye literature. When they’re in trouble it’s always their girl Friday who gets them out of it.
L.B. Jeffries: Well, is she the girl who saves him from the clutches of the seductive show girls and the over passionate daughters of the rich?
Lisa Fremont: The same.
L.B. Jeffries: That’s the one, huh? But he never ends up marrying her, does he? That’s strange.”
—John Michael Hayes (b. 1919)