MthT 430 Notes Chapter 5 Equivalent Definitions

MthT 430 Notes Chapter 5 Equivalent Definitions

What is a definition?

From http://mw1.merriam-webster.com/dictionary/definition

Definition of definition - Merriam-Webster Online Dictionary

Main Entry: def·i·ni·tion

Function: noun

1:: an act of determining ...

2a:: a statement expressing the essential nature of something

2b:: a statement of the meaning of a word or word group or a sign or symbol < dictionary definitions >

Statement 2b seems most appropriate for mathematics.

Think of a Definition as being able to interchange

·: [Definition] Term¹ (What is being defined)

·: [Definition] Description (Details)

Equivalent Definitions of Limit

Definition. (Actual, p. 96)

lim
x ® a
f(x) = L

means: For every e > 0, there is some d > 0 such that, for all x, if 0 < |x -a| < d, then |f(x) - L| < e.
Thus we are able to use interchangeably the phrases

·: ( Definition Term) lim_{x ® a} f(x) = L.

·: (Definition Description) For every e > 0, there is some d > 0 such that, for all x, if 0 < |x -a| < d, then |f(x) - L| < e.

We wish to decide which variations of a definition are "correct" and give an equivalent definition.

Think of a Equivalent Definitions as an If and Only If Theorem .

The phrase "Definition X is equivalent to Definition Y" means you can use interchangeably the phrases

·: [Definition] Term (What is being defined)

·: [Definition] Description X (Details)

·: [Definition] Description Y (Details)

To show that two definitions X and Y for the same Definition Term are equivalent we must show the following:

·: Satisfying Definition Description X Þ Satisfying Definition Description Y.

·: Satisfying Definition Description Y Þ Satisfying Definition Description X.

Now if Definition X is not equivalent to Definition Y for the same Definition Term , then at least one of the following is false:

·: Satisfying Definition Description X Þ Satisfying Definition Description Y.

·: Satisfying Definition Description Y Þ Satisfying Definition Description X.

Interpreting each of the above as a Theorem, the way to show a Theorem is false is to construct a counterexample . A counterexample is an object [construct, ¼] which satisfies the hypotheses of the proposed Theorem, but does not satisfy the conclusion[s] of the proposed Theorem.

Actual Definition of Limit

Definition ACTUAL. (Actual, p. 96)

lim
x ® a
f(x) = L

means: For every e > 0, there is some d > 0 such that, for all x, if 0 < |x -a| < d, then |f(x) - L| < e.
Proposed Variations

For each of the proposed variations AA - OO of the actual (Spivak) definition description of

lim
x ® a

f(x) = L,

decide whether the proposed variation Definition XX is equivalent to Definition Actual. Thus for each you must think about the validity of the the two Theorems:

·: Satisfying Definition Description XX Þ Satisfying Definition Description ACTUAL. If False, there is a counterexample.

·: Satisfying Definition Description ACTUAL Þ Satisfying Definition Description XX. If False, there is a counterexample.

You may construct any counterexample graphically, by formula, or by a precise description.

Definition AA.

lim
x ® a
f(x) = L

means: For every e > 0, there is some d > 0 such that, for all x, 0 < |x -a| < d, and |f(x) - L| < e.

Definition BB.

lim
x ® a
f(x) = L

means: For every e > 0, there is some d > 0 such that, for some x, 0 < |x -a| < d, and |f(x) - L| < e.

Definition CC.

lim
x ® a
f(x) = L

means: For an e > 0, there is some d > 0 such that, for all x, if 0 < |x -a| < d, then |f(x) - L| < e.

Definition DD.

lim
x ® a
f(x) = L

means: For every e > 0, there is some d > 0 such that, for all x, if 0 < |x -a| < d, |f(x) - L| < e.

Definition EE.

lim
x ® a
f(x) = L

means: For any e > 0, there is a d > 0 such that, for all x, if 0 < |x -a| < d, then |f(x) - L| < e.

Definition FF.

lim
x ® a
f(x) = L

means: For any e > 0, there is a d > 0 such that, for all x, 0 < |x -a| < dimplies |f(x) - L| < e.

Definition GG.

lim
x ® a
f(x) = L

means: For any e > 0, there is a d > 0 such that, for all x, |f(x) - L| < eif 0 < |x -a| < d.

Definition HH.

lim
x ® a
f(x) = L

means: For any e > 0, there is a d > 0 such that, for all x, |f(x) - L| < eand 0 < |x -a| < d.

Definition II.

lim
x ® a
f(x) = L

means: For any e > 0, there is a d > 0 such that, for all x, |f(x) - L| < ewhenever 0 < |x -a| < d.

Definition JJ.

lim
x ® a
f(x) = L

means: For every e > 0, there is a d > 0 such that, for all x, |f(x) - L| < efor 0 < |x -a| < d.

Definition KK.

lim
x ® a
f(x) = L

means: For an e > 0, there is a d > 0 such that, for all x, |f(x) - L| < efor 0 < |x -a| < d.

Definition LL.

lim
x ® a
f(x) = L

means: For a d > 0, there is an e > 0 such that, for all x, |f(x) - L| < eprovided that 0 < |x -a| < d.

Definition MM.

lim
x ® a
f(x) = L

means: For all d > 0, there is an e > 0 such that, for all x, |f(x) - L| < efor 0 < |x -a| < d.

Definition NN.

lim
x ® a
f(x) = L

means: For some d > 0, there is an e > 0 such that, for all x, if |f(x) - L| < e, then 0 < |x -a| < d.

Definition OO.

lim
x ® a
f(x) = L

means: For some d > 0, for all e > 0, for all x, |f(x) - L| < e if 0 < |x -a| < d.

Footnotes:

¹I borrow the words Definition Term and Definition Description from the html tags < DT > and < DD > .

File translated from T_EX by T_TH, version 3.77.
On 02 Oct 2007, 12:50.