SUBCRIPTS AND INDEXING

Are used when dealing with large collection of objects especially similar objects.

For example, for n ϵ P and the collection of sets are indexed by subscript:

                Let Dn = { k ϵ Z : k is divisible by n}. List D1, D2, and D3.

                Answer: D1 = { …,  -3, -2, -1, 0 ,1, 2 ,3, ...} = Z

                                  D3= { …, -6, -3, 0, 3, 6 , … }

                                  D5 = { …, -10, -5, 0, 5, 10, … }

                Where 1) D1 is equal to all integers,

                                2) k is a variable (condition),

                                3) We define the condition of the set as { k : properties / predicate / formula }.

SET OPERATIONS

If A and B are two non-zero sets, U is its universal, then we define the operation as following:

a)      UNION

The union of 2 sets A and B is:    { x : x ϵ A ˅  x ϵ B}

Example : If  A = {Aini, Amar, Anis}, B = {Amar, Zati},

                    then A ᴗ B = {Aini, Amar, Anis, Zati}

b)      INTERSECTION

The intersection of 2 sets A and B is:       {  x : x ϵ A ˄ x ϵ B}

Example 1 : If  A = {Aini, Amar, Anis}, B = {Amar, Zati},

                    then A ᴖ  B = { Amar }

Example 2: if : If  A = {Aini, Amar, Anis}, B = {Amar, Zati}, C = {Zain}

then A ᴖ  B ᴖ C = { } because C does not have any similar properties (Amar) with A and B.

c)       COMPLEMENT

¬A = { x : x ϵ A}

 

Example 1:  A = { x : x is shaded}, then

                               ¬A = { x  : x is not shaded}

d)      SYMMETRIC DIFFERENCE

A ⊕ B : {x : ( x ϵ A ˄ x ∉ B) ˅ ( x ϵ B ˄ x ∉ A }

Key point : Take the non-intersect number/properties

Example 1: A = { 0, 1, 2, 3, 4, …,9, 10 },   B = { 0, 2, 4, 6, 8,…,16, 18, 20}

                A ⊕ B = { 1, 3, 5, 7, 9, 12, 14, 16, 18, 20 }

e)      DIFFERENCE / RELATIVE COMPLEMENT

A\B = { x : x ϵ A ˄ x ∉ B}

The difference / relative complement of a set B with respect to set A is not the set of elements which belongs to A but which does not belong to B.

This is denoted by A\B (read as A strip B)

Example 1: A = {4, 5, 7},  B = {2, 4, 5}

                        A\B = {7}

                        B\A = {2}

SET OF ORDERED PAIR

For 2 sets A abd B, for each element a in A and b in B, then their ordered pair is <a,b>.

The set of all ordered pair <a,b> is the product of set A and set B, and can be written as

AxB = { <a,b> :a ϵ A and b ϵ B}

Beware that AxB ≠ Bx A,                               |AxB| = |A| x |B|

Example 1: If A = {1,2,3} and B = {0,1,2,3}, list A and B where,

a) S = {<m,n> ϵ AxB : m+n >3}

b)T= {<m,n> ϵ AxB : m-n =2}

Answer:

a)      S = {<1,3>,<2,2,>, <2,3>,<3,1>,<3,2>,<3,3>}

b)      T = {<2,0>,<3,1>}

Keypoint: Answer for S and T must follow the mentioned properties  in the question and be written in pairs.