Subsets

• Set A is a subset of set B, symbolized A ⊆ B, if an only if all elements of set A are also elements of set B.
  • A = { 1, 2, 3}
  • B = { 2, 4, 6, 8}
  • C = { 1, 2, 3, 4, ... , 10}
  • D = { x | x ∈ N and x ≤ 10}
  • E = { a, b, c}
  • State any relationships between these sets.
  • E ⊄ A
• Set A is a proper subset of set B, symbolized by A ⊂ B, if an only if A ⊆ B but A ≠ B.
  • In other words, B must contain at least one more element than A.
  • Think of ≤ and <
• The empty set is a subset of every set.
• The empty set is a proper subset of all non-empty sets.
• Note sets and elements are different things.
  • A = { 1, 2, 3, ..., 10}
  • 3 ∈ A is correct.
  • 3 ⊂ A is not correct
  • {3} ∈ A is not correct
  • {3} ⊆ A, and {3} ⊂ A are correct
• For a set A, n(A) = k, A has 2^k subsets.
  • {} has 1 subset: {}
  • {a} has two subsets: {} and {a}
  • {a,b} has four subsets: {}, {a}, {b}, {a,b}

• Are the following statements correct
  • {frogs, snakes} ∈ {frogs, lizards, snakes, toads}
  • {frogs, snakes} ⊆ {frogs, lizards, snakes, toads}
  • {frogs, snakes} ⊂ {frogs, lizards, snakes, toads}
  • frogs ⊂ {frogs, lizards, snakes, toads}
  • frogs ∈ {frogs, lizards, snakes, toads}
  • {dogs, frogs} ⊂ {frogs, lizards, snakes, toads}
  • {frogs, lizards, snakes, toads} ⊂ {frogs, lizards, snakes, toads}
  • {frogs, lizards, snakes, toads} ⊆ {frogs, lizards, snakes, toads}
  • Ø ⊆ {frogs, lizards, snakes, toads}
• A pizza parlor offers mushrooms, tomatoes, onions, and sausage as toppings for a plain cheese pizza. How many different types of pizzas can be made?