**Sets and Other Elementary Subjects**

Sets are a collection of things called objects. Objects are all unambiguously defined. In other words, objects have unmistakably clear definitions with one meaning and one interpretation that leads to one conclusion. This may seem convoluted because we are so used to words and phrases having different meanings and whatnot, but not in this case. Look at some examples to get a better idea what it means for objects to be unambiguously defined.

**Objects Not Objects**

Cars Cool cars

Children Nice children

Temperature Comfortable temperature

Baseball players Good baseball players

We need to look at some notation and then I’ll let you know what that notation means:

x∈A x is a member of A

x∉A x is not a member of A

This means that x is a part of group A. Here are some less general examples to emphasize what I mean:

√2∈all real numbers the square root of 2 is a member of all real numbers

√2∉all rational numbers the square root of 2 is not a member of all rational numbers

Sets are groups of common objects and are denoted with curly braces: { }.

a∈{a,b,c}

a∉{x,b,c}

I={1,2,3,…n} I is a set of all positive integers

Sets are often defined by a property such as {a:P}. This means the set of all a for which P holds true. The ':' indicates a property “for which”, and “such that”. Here are some examples:

**Example Statement**

{x:x>2} The set of all numbers greater than 2==the set of x such that x is greater than 2

{x:x^2>9} The set of all numbers such that all their squares are greater than 9

{p/q:p,q integers,q≠0} The set of all rational numbers