Describing Sets

There are two convenient ways to define sets. So far, our sets have been written out in list notation, meaning that each element is written in the set or is implied by a set pattern using "...".

We can also write sets using predicate notation, which uses a variable and a description to introduce the set. The general format of a set written in predicate notation looks like the following:

{VARIABLE | VARIABLE is DESCRIPTION}

Let's look at our old example containing the set of letters and write it out using list notation and predicate notation.

• Letters = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}
• Letters = {x | x is a letter of the English alphabet}
  • This is read as "the set of x such that x is a letter of the English alphabet".

Here's another example using primary colors.

• P = {red, yellow, blue}
• P = {x | x is a primary color}
  • This is read as "the set of x such that x is a primary color".

We say that two sets are equivalent iff they contain exactly the same elements. By definition, any set written in list notation is equivalent to the same set written in predicate notation.

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