Math 141 Lecture Notes



Math 141 Lecture Notes

Section 6.1 Sets and Set Operations

Set Terminology and Notation

A set is a well-defined collection of objects. The objects are called the elements and are usually denoted by lowercase letters a, b, c, …; the sets themselves are usually denoted by uppercase letters A, B, C, ….

The elements of a set may be displayed using roster notation by listing each element between braces (ex. B = { a, b, c, . . . , z }) or by roster notation by giving a rule that describes the definite property or properties an object x must satisfy to qualify for membership in the set (ex. B = {x | x is a letter of the English alphabet}.

Two sets A and B are equal iff they have exactly the same elements.

Example 1: Let A, B, and C be the sets

A = {a, e, i, o, u}

B = {a, i, o, e, u}

C = {a, e, i, o}

A = B since they both contain exactly the same elements. A ( C since u ( A but u ( C. Also, B ( C.

If every element of a set A is also an element of a set B, then we say that A is a subset of B (A ( B).

Example 2: Let A, B, C, and C be the sets

A = {a, e, i, o, u}

B = {a, i, o, e, u}

C = {a, e, i, o}

D = {a, e, i, o, x}

C ( since every element of C is also an element of B. D ( A since x ( D but x ( D.

If A and B are sets such that A ( B but A ( B, then A is a proper subset of B (A ( B).

Example 3: Let A = {1, 2, 3, 4, 5, 6} and B = {2, 3, 6}.

B is a proper subset of A since (1) B ( A, and (2) there exists at least one element in A that is not in B.

The set that contains no elements is called the empty set and is denoted by (.

Example 4: List all subsets of the set A = {a, b, c}.

The universal set is the set of all elements of interest in a particular discussion. It is the largest in the sense that all sets considered in the discussion of the problem are subsets of the universal set.

Example 5: Give the universal set for each situation:

a) Determine the ratio of female to male students in a college.

b) Determine the ratio of female to male students in the business department of the college.

Venn diagrams are used as a visual representation of sets. The universal set U is represented by a rectangle, and subsets of U are represented by regions lying inside the rectangle.

Example 6: Use Venn diagrams to illustrate the following statements:

a. The sets A and B are equal.

b. The set A is a proper subset of the set B.

c. The sets A and B are not subsets of each other.

Set Operations

The union of sets A and B (A ( B) is the set of all elements that belong to either A or B or both. A ( B = {x | x ( A or x ( B or both}

[pic]

Example 7: If A = {a, b, c} and B = {a, c, d}, find A ( B. Draw a Venn diagram to illustrate the problem.

The intersection of sets A and B (A ( B) is the set of all elements in common with the sets A and B. A ( B = {x | x ( A and x ( B}

[pic]

Example 8: If A = {a, b, c} and B = {a, c, d}, find A ( B. Draw a Venn diagram to illustrate the problem.

Example 9: If A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10}, find A ( B. Draw a Venn diagram to illustrate the problem.

Two sets are disjoint is they have no elements in common, that is, if A ( B = (.

If U is a universal set and is a subset of U, then the set of all elements in U that are not in A is called the complement of A and is denoted [pic].

[pic] = {x | x ( U, x ( A}

[pic]

Example 10: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}, find [pic]. Illustrate with a Venn diagram.

If U is a universal set and A is a subset of U, then

a. [pic] ( b. (C = U c. [pic] d. [pic] = ( e. [pic] = (

Let U be a universal set. If A, B, and C are arbitrary subsets of U, then

A ( B = B ( A Commutative law for union

A ( B = B ( A Commutative law for intersection

A ( (B ( C) = (A ( B) ( C Associative law for union

A ( (B ( C) = A ( (B ( C) Associative law for intersection

A ( (B ( C) = (A ( B) ( (A ( C) Distributive law for union

A ( (B ( C) = (A ( B) ( (A ( C) Distributive law for intersection

De Morgan’s Laws

Let A and B be sets. Then, [pic] and [pic]

Example 11: Using Venn diagrams, show that [pic]

Example 12: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 20}, A = {1, 2, 4, 8, 9}, and B = {3, 4, 5, 6, 8}, show that [pic].

Example 13: Let U denote the set of all students in Math 141 and

A = {x ( U | x likes to play forty-two}

B = {x ( U | x loves math}

C = {x ( U | x lives in a dorm}

Find an expression in terms of A, B, and C for each of the following sets:

a. The set of Math 141 students with at least one of the given characteristics

b. The set of Math 141 students with exactly one of the given characteristics

c. The set of Math 141 students live in a dorm and love math but do not like to play forty-two

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