let X = { 1 , 2 , 3 ,4 } and Y = { 3 , 4 , 5 , 6 , 7 , 8 } then find X ∪ Y , X ∩ Y , X - Y , Y - X

**1 - (X ∪ Y ) union**

∪ this symbol is called union means all elements in X and all elements in Y and no need to repeat the elements

then X ∪ Y = { all elements in X and all elements in Y don't repeat the elements } ={1, 2, 3 ,4 ,5,6, 7 ,8 }

**2- (X ∩ Y) intersection**

∩ this symbol is called intersection means the common elements between X & Y

then X ∩ Y = { the common elements between X & Y}= { 3 , 4 }

If the intersection of two sets

{1, 2} ∩ {3, 4} = ∅.

If the intersection of two sets

*A*and*B*is empty, that is they have no elements in common, then they are said to be**disjoint**, denoted:*A*∩*B*= ∅. For example the sets {1, 2} and {3, 4} are disjoint, written{1, 2} ∩ {3, 4} = ∅.

**3 - (X - Y )difference**

- means elements in X but not in Y (notes that X is first )

then X - Y = {elements in X but not in Y }= {1 , 2}

**4-( Y - X ) difference**

- elements in Y but not in X (note that Y is first)

then Y - X = {elements in Y but not in X}={5, 6 ,7 ,8}

ex : A = { 5 , 8 , 9 } & B = { 4 , 6 ,8 , 9 , 10 , 11 }

then

if not all elements of A not found in B then A is not a subset of B (A B)

-----------------------------------------------------------------

let A & B are to sets then A is a subset of B **(A ⊂ B)**if all elements of A are found in Bex : A = { 5 , 8 , 9 } & B = { 4 , 6 ,8 , 9 , 10 , 11 }

then

**A⊂B**because all elements of A (5, 8 ,9 ) are found in Bif not all elements of A not found in B then A is not a subset of B (A B)

-----------------------------------------------------------------

**Other basic concepts**

**universal set**(U) : is a set which contains all objects such as if A = { 5 , 8 , 9 } then U = N (natural number set)**Complement of set A(***A*^{c})^{ }: it is the set of all number in (U) not in (A)**Symmetric difference**of sets*A*and*B is set of elements in A not in B and elements in B not in A*For instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4}**Power set**of a set A is the set whose members are all possible subsets of*A*. For example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } .

**some solved problems :**

*1) zero ∈*{ X - 3 , 4 } therefore X = ....................................because zero

*∈*{ X - 3 , 4 } then X should equal 4 or X - 3 but zero ≠ 4 then zero shouldequal X - 3

then X - 3 = zero then X = 3

**2) { 3 , X + 1} ⊂ {4 , 3 } therefore X = ...................................**

BECAUSE 3 = 3 THEN X + 1 = 4 then X = 3

**3) { 2,1}∪ { 3 , X - 1 } = {1,2,3,4}THEN X = ...................................**

1 = 1 , 2=2 and 3 =3 then X -1 = 4 then X = 5

**4) {2X , 8 }∩ {5,12}= {12}then X = ....................................**

the common element is 12 then 12 should be in the two sets then12 = 2X because 8 ≠ 12 then

X = 6

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