DS-notes
Lado Sitchinava
&1 Sets
1.if all element is set 𝐴 and set 𝐵 are equal then we write 𝐴 = 𝐵
2.Intersection of sets are set of all the simlar elements set 𝐴 and set 𝐵 have. we write it as 𝐴 ∩ 𝐵
3.Difference of Set 𝐴 and Set 𝐵 is set of all elements that are in 𝐴 but cannot be found in set 𝐵.
formal definition 𝐴 \ 𝐵
4.Cartesian product of Set 𝐴 and set 𝐵 is set of A’s and B’s elements concatenated. For example:
𝐴 = {1, 2}, 𝐵 = {3, 4}
𝐴𝑋𝐵 = {(1, 3), (1, 4), (2, 3), (2, 4)}
5.power set of set 𝐴 for example is set of all subsets of 𝐴. number of subsetst of 𝐴 is 2𝑛 .n is number
of elements in set 𝐴.
&2 Mathematical induction
I will skip this because its hell easy
&3 Comutativity,associativity and Distributivity
1.Comutativity: 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴, 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴
2.Associativity: (𝐴 ∪ 𝐵) ∪ 𝐶 = 𝐴 ∪ (𝐵 ∪ 𝐶), (𝐴 ∩ 𝐵) ∩ 𝐶 = 𝐴 ∩ (𝐵 ∩ 𝐶).
3.Distributivity:𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶),𝐴 ∩ (𝑏 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶).
De Morgans’s Laws : 𝐶 \ (𝐴 ∩ 𝐵) = (𝐶 \ 𝐴) ∪ (𝐶 \ 𝐵).
&4 Function and mapping
1.𝑓 : 𝑋 → 𝑌 , here X is domain(განსსაზღვრის არე),Y is co-domain(მნიშვნელობათა არე)
2.Identity function: Let X be a set
𝑋→𝑋
id𝑥 : {𝑥→𝑥 this is called identity function on X
2.Let X and Y be sets.
𝑋×𝑌 →𝑋
𝜋1 : {
(𝑥, 𝑦) → 𝑥
this is called projection on first factor, which means that every (x,y) paris provides output of x.
3.[𝑥] is largest integer smaller or equal to x(მთელი ნაწილი)
4.{x} is smallest integer larger or equal to x(წილადი ნაწილი)
&5 Injectivity,surjectivity, bijectivity
1.definition of injectivity: function’s output is unique in every case for example: one domain should
relate to one co-domain but other domain can not be connected to same co-doamin as first domain
Lado Sitchinava
&1 Sets
1.if all element is set 𝐴 and set 𝐵 are equal then we write 𝐴 = 𝐵
2.Intersection of sets are set of all the simlar elements set 𝐴 and set 𝐵 have. we write it as 𝐴 ∩ 𝐵
3.Difference of Set 𝐴 and Set 𝐵 is set of all elements that are in 𝐴 but cannot be found in set 𝐵.
formal definition 𝐴 \ 𝐵
4.Cartesian product of Set 𝐴 and set 𝐵 is set of A’s and B’s elements concatenated. For example:
𝐴 = {1, 2}, 𝐵 = {3, 4}
𝐴𝑋𝐵 = {(1, 3), (1, 4), (2, 3), (2, 4)}
5.power set of set 𝐴 for example is set of all subsets of 𝐴. number of subsetst of 𝐴 is 2𝑛 .n is number
of elements in set 𝐴.
&2 Mathematical induction
I will skip this because its hell easy
&3 Comutativity,associativity and Distributivity
1.Comutativity: 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴, 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴
2.Associativity: (𝐴 ∪ 𝐵) ∪ 𝐶 = 𝐴 ∪ (𝐵 ∪ 𝐶), (𝐴 ∩ 𝐵) ∩ 𝐶 = 𝐴 ∩ (𝐵 ∩ 𝐶).
3.Distributivity:𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶),𝐴 ∩ (𝑏 ∪ 𝐶) = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶).
De Morgans’s Laws : 𝐶 \ (𝐴 ∩ 𝐵) = (𝐶 \ 𝐴) ∪ (𝐶 \ 𝐵).
&4 Function and mapping
1.𝑓 : 𝑋 → 𝑌 , here X is domain(განსსაზღვრის არე),Y is co-domain(მნიშვნელობათა არე)
2.Identity function: Let X be a set
𝑋→𝑋
id𝑥 : {𝑥→𝑥 this is called identity function on X
2.Let X and Y be sets.
𝑋×𝑌 →𝑋
𝜋1 : {
(𝑥, 𝑦) → 𝑥
this is called projection on first factor, which means that every (x,y) paris provides output of x.
3.[𝑥] is largest integer smaller or equal to x(მთელი ნაწილი)
4.{x} is smallest integer larger or equal to x(წილადი ნაწილი)
&5 Injectivity,surjectivity, bijectivity
1.definition of injectivity: function’s output is unique in every case for example: one domain should
relate to one co-domain but other domain can not be connected to same co-doamin as first domain
