Direct Proof

Pronunciation: /dɪˈrɛkt ˌpruf/ Explain

A direct proof is a mathematical proof that uses axioms, definitions, and previously proved theorems without making any further assumptions.[1]

Example: Proof that the sum of two even integers is even

Consider two arbitrary even integers x and y. The definition of an even integer is: An even integer can be written as 2a where a is an integer. We will show that there exists an integer c such that x + y = 2c. If x + y = 2c, then the sum of x and y is even.

Using the definition of an even integer, the two integers x and y can be rewritten as x = 2a and y = 2b where a and b are integers. Then the sum x + y can be written as 2a + 2b.

Using the distributive property of multiplication, 2a + 2b can be written as 2(a + b). Since a and b are integers, and the set of integers is closed with respect to addition, there exists an integer c = a + b. 2(a + b) can then be rewritten as 2(c) = 2c. Since c is an integer, the expression 2c matches the definition of an even number. QED.

Proof in two column form

Consider two arbitrary even integers x and y. Initial assertion
An even integer can be written as 2a where a is an integer. Definition of an even integer.
We will show that there exists an integer c such that x + y = 2c. Claim
x and y can be rewritten as x = 2a and y = 2b where a and b are integers. Apply the definition of an even integer.
The sum x + y can be written as 2a + 2b. Substitute 2a for x and 2b for y.
2a + 2b can be written as 2(a + b). Apply the distributive property of multiplication.
There exists an integer c = a + b. Apply the closure property of integers and addition.
2(a + b) can then be rewritten as 2(c) = 2c. Substitute c for a + b.
QED. The proof is complete.

References

  1. Cupillari, Antonella. Nuts and Bolts of Proof: An Introduction to Mathematical Proofs. 3rd edition. pp 12-17. Academic Press. August 15, 2005. Last Accessed 8/6/2018. Buy the book
  2. Ethan D. Bloch. Proofs and Fundamentals: A First Course in Abstract Mathematics. 1st Edition. pp 63-66. Birkhäuser Boston. April 20, 2000. Last Accessed 8/6/2018. Buy the book

More Information

  • SparkNotes Editors. SparkNote on Geometric Proofs. sparknotes.com. SparkNotes LLC. 11/22/2009. http://www.sparknotes.com/math/geometry3/geometricproofs/summary.html.

Cite this article as:

McAdams, David E. Direct Proof. 7/5/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/directproof.html.

Revision History

7/4/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
1/23/2010: Added "References". (McAdams, David E.)
11/21/2009: Added "More Information". (McAdams, David E.)
3/10/2008: Added vocabulary hot links. (McAdams, David E.)
2/4/2008: Fixed formatting problem. (McAdams, David E.)
7/13/2007: Initial version. (McAdams, David E.)

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