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Section 2.2: Subsets. Objectives Recognize subsets and use the notation  . Recognize proper subsets and use the notation . Determine the number of subsets of a set. Subsets.

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section 2 2 subsets
Section 2.2: Subsets

Objectives

  • Recognize subsets and use the notation .
  • Recognize proper subsets and use the notation .
  • Determine the number of subsets of a set.
subsets
Subsets
  • Set A is a subset of set B, symbolized A ⊆ B, if and only if all elements of set A are also elements of set B.
  • The symbol A ⊆ B indicates that “set A is a subset of set B.”
  • The symbol A ⊈ B set A is not a subset of set B.
  • To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B.
determining subsets
Determining Subsets

Example:

A = { 3, 5, 6, 8 }

B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

  • Determine whether set A is a subset of set B.

Solution:

  • Determine whether set B is a subset of set A.

Solution:

proper subset
Proper Subset
  • Set A is a proper subset of set B, symbolized A ⊂ B, if and only if all of the elements of set A are elements of set B and set A ≠ B.
  • (that is, set B must contain at least one element not is set A).
determining proper subsets
Determining Proper Subsets

Example:

A = { dog, cat }

B = { dog, cat, bird, fish }

  • Determine whether set A is a proper subset of set B.

Solution:

  • Determine whether set B is a proper subset of set A.

Solution:

determining proper subsets1
Determining Proper Subsets

Example:

Determine whether set A is a proper subset of set B.

A = { dog, bird, fish, cat }

B = { dog, cat, bird, fish }

Solution:

number of distinct subsets
Number of Distinct Subsets

The number of distinct subsets of a finite set A is 2n, where n is the number of elements in set A.

Example:

  • Determine the number of distinct subsets for the given set {t, a, p, e}.
  • List all the distinct subsets for the given set {t, a, p, e}.
number of distinct proper subsets
Number of Distinct Proper Subsets

The number of distinct proper subsets of a finite set A is 2n– 1, where n is the number of elements in set A.

Example:

Determine the number of distinct proper subsets for the given set{t, a, p, e}.

Solution: