1.1 Sets and Subsets

A set is a well defined collection of objects A collection of beanie babies A collection of hats An Element (∈) is one of the objects in a set A = {1, 2, 3} 1 ∈ A 2 ∈ A 3 ∈ A 4 ∉ A

We specify a property when it is difficult to list all elements. We write this in the form {x|P(x)} (x such that P of x) A = {1,2,3} could be written as: {x|x is a positive integer less than 4}

Order in a set does not matter {1,2,3} {3,2,1} Special sets: ℤ(zeta) integers {..., −3, −2, −1, 0, 1, 2, 3, ...} ℕ(nu) natural numbers { 0, 1, 2, 3, ...} or { 1, 2, 3, ...}. ℚ Rational numbers integers, fractions and decimals ℝ Real numbers Rational and Irrational numbers (square root ) {}Empty set

Equality Two sets A and B are equal if they have the same elements: If A= {Java, Pascal, c++} and B = {Pascal, c++, Java} then A=B

Subsets ⊆ is contained in A ⊆ B A={1,2,3,4} B = {1,2,5,6} A is not contained in B

Venn Diagrams The Venn diagram shows relationships between sets. of what we are talking about A = a set of beanie babies B = a set of hats The intersection ∩ of A and B, written A ∩B are beanie babies with hats.

The Power Set of A , written P(A), is the set of all possible subsets of A. Note there are 3 elements 23 = 8 There are 8 possible subsets P(A) = {}, {1}, {2}, {3}, {1,2},{1,3},{2,3},{1,2,3}