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Note that if we don't have the Extremal Clause,  0.5, 1.5, 2.5, ... can be included in N, which is not what we want as the set of natural numbers.

Example 2. Definition of the Set of Nonnegative Even Numbers NE

The set NE is the set that satisfies the following three clauses:

Basis Clause: 0 ∈ NE

Inductive Clause: For any element x in NE, x + 2 is in NE.

Extremal Clause: Nothing is in NE unless it is obtained from the Basis and Inductive Clauses.

Example 3. Definition of the Set of Even Integers EI

The set EI is the set that satisfies the following three clauses:

Basis Clause: 0 ∈ EI

Inductive Clause: For any element x in EI, x + 2, and x - 2 are in EI.

Extremal Clause: Nothing is in EI unless it is obtained from the Basis and Inductive Clauses.

Example 4. Definition of the Set of Strings S over the alphabet {a,b} excepting empty string. This is the set of strings consisting of a's and b's such as abbab, bbabaa, etc.

The set S is the set that satisfies the following three clauses:

Basis Clause: a ∈ S, and b ∈ S.

Inductive Clause: For any element x in S, ax ∈ S, and bx ∈ S.

Here ax means the concatenation of a with x.

Extremal Clause: Nothing is in S unless it is obtained from the Basis and Inductive Clauses.

Tips for recursively defining a set:

For the "Basis Clause", try simplest elements in the set such as smallest numbers (0, or 1), simplest expressions, or shortest strings. Then see how other elements can be obtained from them, and generalize that generation process for the "Inductive Clause".

The set of propositions (propositional forms) can also be defined recursively.

Generalized set operations

As we saw earlier, union, intersection and Cartesian product of sets are associative. For example (A ∪ B) ∪ C = A ∪ (B ∪ C)

To denote either of these we often use A ∪ B ∪ C.

This can be generalized for the union of any finite number of sets as A1 ∪ A2 ∪.... ∪ An.

which we write as

       i = 1 n A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {}

This generalized union of sets can be rigorously defined as follows:

Definition ( i = 1 n A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} ):

Basis Clause: For n = 1, i = 1 n A i = A 1 size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } =A rSub { size 8{1} } } {} .

Inductive Clause:   i = 1 n + 1 A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n+1} } A rSub { size 8{i} } } {} = i = 1 n A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} ∪ An+1

Similarly the generalized intersection i = 1 n A i size 12{ intersection rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} and generalized Cartesian product i = 1 n A i size 12{ times rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} can be defined.

Based on these definitions, De Morgan's law on set union and intersection can also be generalized as follows:

Theorem (Generalized De Morgan)

i = 1 n A i ¯ = i = 1 n A i ¯ size 12{ {overline { union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } }} = intersection rSub { size 8{i=1} } rSup { size 8{n} } {overline {A rSub { size 8{i} } }} } {} ,     and

i = 1 n A i ¯ = i = 1 n A i ¯ size 12{ {overline { intersection rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } }} = union rSub { size 8{i=1} } rSup { size 8{n} } {overline {A rSub { size 8{i} } }} } {}

Proof: These can be proven by induction on n and are left as an exercise.

Recursive definition of function

Some functions can also be defined recursively.

Condition: The domain of the function you wish to define recursively must be a set defined recursively.

How to define function recursively: First the values of the function for the basis elements of the domain are specified. Then the value of the function at an element, say x, of the domain is defined using its value at the parent(s) of the element x.

A few examples are given below.

They are all on functions from integer to integer except the last one.

Example 5: The function f(n) = n! for natural numbers n can be defined recursively as follows:

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
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