SET

If A denoted a set and if x is an element we shall write,

x ∈A

as an abbreveation for the statement that x is an element of A or that x is a member of A or that x belong to A or that the set a contain the element X or that x is in A. If x is an element that does not belong to A, we shall write 

x ∉ A

If A and B are set such as x ∈A implies that x ∈B (that is every element of A is also an element of B), then we shall say that A is contained in B or that B contains A, or that A is a subset of B and we shall write.

A⊆B or B⊇A

Definition, two sets A and B are equal if they contain the same element. If the sets A and B are equal, we write A=B

Examples:

1. The set {x∈N : x²‒3x+2=0} consists of those natural numbers satisfying the stated equation since the only solutions of the quadratic equation x²‒3x+2=0 are x=1 and x=2, instead of writing the above expression we ordinarily denoted this set by {1,2}, thereby l;isting the elements of the set.

2. Sometimes of formula can be used to abbreviate the description of a set. For example, the set of all even natural numbers could be denoted by {2x : x∈N}, instead of the more cumbersome {y∈N : y=2x, x∈N}

3. The set {x∈N : 6<x<9} can be written explicitly as {7,8}, thereby exhibiting the elements of the set. Of course, there are many other possible description of this set, for example: 

   {x∈N : 40<x²<80} 

   {x∈N : x²‒15x+56=0} 

   {7+x : x=0 or x=1}