DEFINITIONS
Even Integer |
An integer x is even if there is an integer k such that x = 2k. |
Odd Integer |
An integer x is odd if there is an integer k such that x = 2k+1. |
Parity |
Whether the number is odd or even |
Divides |
An integer x divides an integer y if and only if x ≠ 0 and y = kx, for some integer k. Denoted x|y. If x does not divide y, then that fact is denoted x ∤ y. If x divides y, then y is said to be a multiple of x, and x is a factor or divisor of y. |
Prime |
An integer n is prime if and only if n > 1, and the only positive integers that divide n are 1 and n. |
Composite |
An integer n is composite if and only if n > 1, and there is an integer m such that 1 < m < n and m divides n. |
Rational |
A number r is rational if there exist integers x and y such that y ≠ 0 and r = x/y. |
ZERO |
0 is rational. For example if x = 0 and y = 1, then y ≠ 0 and x/y = 0/1 = 0. |
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METHOD DEFINITIONS
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constructive proof of existence A proof that shows that an existential statement is true.
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proof by exhaustion |
Allowed assumptions in proofs
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The rules of algebra. For example if x, y, and z are real numbers and x = y, then x+z = y+z.
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The set of integers is closed under addition, multiplication, and subtraction. n other words, sums, products, and differences of integers are also integers.
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Every integer is either even or odd. This fact is proven elsewhere in the material.
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If x is an integer, there is no integer between x and x+1. In particular, there is no integer between 0 and 1.
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The relative order of any two real numbers. For example 1/2 < 1 or 4.2 ≥ 3.7.
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The square of any real number is greater than or equal to 0. This fact is proven in a later exercise.
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Common keywords and phrases in proofs
Thus, therefore then, hence, it follows that
A statement that follows from the previous statement(s)
ex. n and m are integers. Therefore, n+m is also an integer.
Let, suppose
Introduce a new variable
ex. "Let x be a positive integer" "Suppose that x is a positive integer"
Since
If a statement depends on a fact that appeared earlier in the proof or in the assumptions of the theorem, it can be helpful to remind the reader of that fact before the statement.
ex. "Since x > 0 and y > z, then xy > xz."
By definition
A fact that is known because of a definition
ex. "The integer m is even. By definition, m = 2k for some integer k."
By assumption
A fact that is known because of an assumption
ex. "By assumption, x is positive. Therefore x > 0."
"gives" and "yields"
useful to say that one equation or inequality follows from another
provides clarity to justify algebraic steps
*ex. Multiplying both sides of the inequality x > y by 2 gives 2x > 2y.
Substituting m = 2k into m2 yields (2k)2*
Since z > 0, we can multiply both sides of the inequality x > y by z to get xz > yz. |
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