Direct Proof

Pronunciation: /dɪˈrɛkt ˌpruf/ ?

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. (Accessed: 2010-01-11).
  2. Ethan D. Bloch. Proofs and Fundamentals: A First Course in Abstract Mathematics, 1st Edition, pp 63-66. Birkhäuser Boston, April 20, 2000. (Accessed: 2010-01-23).

More Information

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

Printed Resources

Cite this article as:


Direct Proof. 2010-01-23. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/directproof.html.

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2010-01-23: Added "References" (McAdams, David.)
2009-11-21: Added "More Information" (McAdams, David.)
2008-03-10: Added vocabulary hot links (McAdams, David.)
2008-02-04: Fixed formatting problem (McAdams, David.)
2007-07-13: Initial version (McAdams, David.)

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