Algebra of the sets
Sets
1) What is together:
Everyone of which is a collection of (objects/individuals) is said element of together.
2) What is preaching:
It is an affirmation that it can be true or false.
3) Which are the ways in order to define the elements pertaining to together:
to) Exaustively belt fasteners are enclosed between parenthesis the single elements separate to you from virgole.
b) {x | P(x)} through preaching that it characterizes the members of together.
4) Enounce the characteristics of the N entirety:
The entire ones are the natural numbers that is positi to you with the 0 included.
5) Enounce the characteristics of the Z entirety:
They are the entire numbers positi and denied to you to you.
6) Enounce the characteristics of the Q entirety:
They are the numbers rations given them from the relationship between 2 entire numbers.
7) As the belongings of one individual x to one with I are denoted:
x ? I
8) That difference is between the symbols ? and :
? refers to the belongings of an element to together
refers to the belongings of with to an other together.
9) When 2 sets they are said equal:
When the same elements contain exactly. It must therefore be verified both:
To B e B To
10) When To it is said sottoset of B, To ? B :
When all the elements of To are comprised in B but not all the elements of B are comprised in To.
11) What is a sottoset just:
A sottoset is had just when all the elements of with To are comprised in the B entirety and it is not with empty.
12) What is with of the parts of To:
It is an entirety constituted from all the possible ones sottosets of To, also is said indicated entirety power and with 2|To| .
13) What is the meaning for cardinalità of with To:
It is the number of the elements that constitute with To, is denoted with |To|.
14) Which are the eseguibili operations on the sets :
Union, intersection, difference.
15) Disegnare in green the union of 2 sets , To ? B:
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16) Disegnare in green the intersection of two sets , To ? B:
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17) Disegnare in green the difference of with of left from with of right, To \ B:
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18) What is the meaning for dominion of together:
The entirety reaches the elements from a dominion carrying out a selection to the aim to characterize the elements that answer to the characteristics of with, therefore the entirety is always a sottoset of the dominion.
19) What is the meaning for complement of with To in the U dominion:
It is with of the individuals of the U dominion that do not make part of with To.
20) Disegnare in green the complement to To in the U dominion:
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21) Which are the characteristics of the complement of with To:
to) To joined to its complement it is the dominion.
b) the intersection between To and its complement are with empty.
22) Enounce the two Laws of De Morgan using the diagrams of Venn:
C(To ? B) = C(A) ? C(B) C(To ? B) = C(A) ? C(B)
23) What is one tidy brace:
It is an object formed from an element to ? C$r-at and from an element b ? B taken in the order,
24) What is the produced entirety cartesian To x B:
Are the entirety formed from all the tidy braces < a,b > con to ? A e b ? B.
Relations
25) What is the meaning for relation between 2 sets :
B and data 2 sets , say relation s between To and B whichever sottoset of the cartesian product To x B, in fact one relation completely is defined when it is settled down with of contained braces < a,b > in with To x B which satisfy the relation.
26) which property they can enjoy the relations:
to) Reflectivity
b) Symmetry
c) Asymmetry
d) Transitività
27) When a relation is reflecting:
It must happen that " to ? To, the brace < a,a > ? s , it is that is contained in the entirety characterized from the relation.
28) When a relation is symmetrical:
They must make part of the entirety characterized from the relation s , is the braces < a,b > that the correspondents braces < b,a >.
29) When a relation is antisymmetric:
They must make part of the entirety characterized from the relation s , is the braces < a,b > that the correspondents braces < b,a > on condition that only are b = to.
30) When a relation is transitiva:
When it happens that if to is in relation with b and b it is in relation with c then also to is in relation with c.
31) Which are the 2 main types of relations:
to) order relations.
b) equivalence relations.
32) Which are the characteristics of one order relation:
Reflectivity, asymmetry , transitività.
33) When an order relation is total:
If all the braces make part of an order relation that is if for all they are worth the reflectivity property, asymmetry , transitività.
34) As it comes defined one relation of partial order:
All are partial relations of order the order relations that are not totals.
35) That grafi is it associates you to the partial relations of order and which instead to the order relations totals:
The order relations totals are in kind characterized from grafi linear in which it is known well which object precedes following. The partial relations of order instead are characterized from grafi to tree.
36) Which are the characteristics of one equivalence relation:
Reflecting, symmetrical, transitiva.
37) What is the meaning for partition generated from the equivalence relation:
The equivalence relations create separate classes that do not have some intersection and whose sum is the total.
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38) What is the meaning for equivalence class:
A class agrees that collects the objects that to the ends of the relation in issue are equivalents, tasks to the case of the relation of equivalence between the straight ones of the plan, will be infinite various classes of equivalence, ciascuna characterized from a various direction, and to the straight inside of ognuna of they parallels to the characteristic direction of that particular class of equivalence will be the ¥.
39) What is the meaning for entirety quotient or with of the classes of the rests K module:
It is with of the classes of equivalence of the relation regarding with on which they are calculated. Customarily draft of a smaller entirety regarding with of departure, just for the characteristic of the relations of equivalence to create classes of equal objects to the ends of the relation. In the case of the relation that alloy the value of the entire ones positi you to the rest of their division for 2, the entirety quotient is constituted from the class 1 containing all the uneven numbers and having therefore rest 1e from the containing all equal numbers and having class the 0 therefore rest 1.
Functions
40) That difference is between relation and one function:
The function is a particular type of relation which preclude the possibility that a same element of with of existence is in relation with 2 various elements of the image, like saying that on a remote control, a same push-button cannot characterize 2 various channels.
41) What is the dominion of one function:
It is with to the inside of which they come selects the elements to you that through the function correspond to some element of the codominio.
42) What is codominio of one the function:
It is with to the inside of which they make also it leave the elements that through the function correspond to some element of with of existence of the function in the dominion.
43) What is with of existence of one function:
It is with of the elements of the dominion which have a correspondent in the codominio through the function.
44) What is the image of one function:
The image of a function between To and B is with of the values of B that correspond to some value to ? To.
45) When a function is iniettiva:
A function is iniettiva if to every element of the image a single element of the dominion corresponds.
46) When a function is suriettiva:
If the image is all the codominio
47) When a function is bigettiva:
When it is is suriettiva iniettiva that.
Calculation of the proposizioni
48) What is one proposition :
Straight affirmation from a verbo is one.
49) What is connects to you logical:
They are of operating having the scope to arrange means you more prepositions, they are:
? AND and
? OR or, or
® IMPLICA then
º EQUIVALENZA is equal
? NOT not
50) What is the calculation of the proposizioni:
It is a calculation, making part of mathematical logic, than it is taken care of the control of the correctness of a reasoning. It characterizes in the phrases the proposizioni and you connect to you logical and is in a position to extracting a table of truth of the date phrase based on the value assumed from the proposizioni that constitute it.
51) Which are the rules for the formation of one corrected composed proposition :
to) Every proposition formula (p) is one
b) If p it is a formula then the not if present must precede (?p)
c) If p1 and p2 they are formulas, then is formulas also following:
p1 ? p2 true when they are both true ones
p1 ? p2 true when at least one is true
p1 ® p2 is false only when p1 it is true and p2 it is false
p1 º p2 true when they are or both true or both false ones
52) Which it is the meant one of the passage from the calculation of the proposizioni to the semantic calculation:
The calculation of the proposizioni does not have sense if it is not rapportato to the world, is necessary a context that is regarding which to assert that one given proposition is true or rather is false.
53) When 2 formulas are equivalents:
When the calculation of the semantic one of the 1ª is equal to the calculation of the semantic one of the 2ª, an algorithm becomes therefore necessary which it concurs to characterize a formula representative of all the formulas whose calculation of the semantic one is equal, forming in such a way one equivalence class.
54) What is the normal congiuntiva shape:
It is a way to reduce a complex proposition to the aim to characterize a formula that represents all one equivalence class. In the normal congiuntiva shape, such final formula possesses which connecting logical the solo ? and.
The final formula will introduce with this aspect: f = f1 ? f2 ? f3 , is a shape used mostly for the expert systems.
55) Which property are usable for the reduction of one formula:
to) sem (to º b) = sem((to ® b) ? (b ® a)) has equivalence quando to implies b and to contempo the b it implies to
b) sem (to ® b) = sem(? to ? b)
c) sem (a) = sem( ? ? to)
You read of De Morgan
d) sem ( ? (to ? b)) = sem( ? to ? ? b)
and) sem ( ? (to ? b)) = sem( ? to ? ? b)
distributive Laws
f) sem (to ? (b ? c) = sem((a ? b) ? (to ? c))
g) sem (to ? (b ? c) = sem((a ? b) ? (to ? c))
56) Which it is the algorithm of reduction to the normal congiuntiva shape:
A) Eliminare the symbols º , ® using property 1 and 2
to repeat these steps alternatively until is necessary:
B) Eliminare the double negations using the 3
C) Utilizzare the laws of De Morgan in order to remove the negations of conjunctions or disjunctions
D) Applicare distributive laws 6 and 7
Applying this algorithm is reached the disjunctive shape or congiuntiva normal school, in the course of the development, to make attention to eliminate also formulas that son always true like (to ? ? to).
57) What is the normal disjunctive shape:
It is a way to reduce a complex proposition to the aim to characterize a formula that represents all one equivalence class. In the normal disjunctive shape, such final formula possesses which connecting logical the solo ? or.
The final formula will introduce with this aspect: f = f1 ? f2 ? f3 is a used shape mostly in order to reduce the digital circuits elettronici them.