Lesson 1.1 Sets and Subsets pp. 2-5

Objectives: 1. To use correct notation to express sets, elements, and subsets. 2. To identify equivalent sets by using one-to-one correspondences. 3. To distinguish equal sets from equivalent sets.

What is Geometry? Our word geometry comes from two Greek words and means “earth measure”

Most importantly, geometry trains your mind to think logically and clearly even in nonmathematical situations.

A = {compass, pencils, ruler, protractor, calculator} A set is a group or collection of objects. Sets are denoted by set braces and are named with a capital letter. A = {compass, pencils, ruler, protractor, calculator}

The objects of the set are elements or members of the set. If A = {compass, pencils, ruler, protractor, calculator} then compass  A.

There are two ways to describe elements of a set. 1. The list method ex. A = {compass, pencils, ruler, protractor, calculator} 2. The description method (set- builder notation) ex. A = {x|x is geometric tool}

The general form of set-builder notation is {x|x is … }, where x is an arbitrary element of the set, and the vertical line, |, indicates the words such that. The vertical line is followed by a description of a representative element of the set.

To symbolize that an object is an element of a set, you can use the symbol . This symbol means “is an element of.”

C = {comb, brush, toothbrush, paste, soap, razor, cloth} EXAMPLE 1 Use the list method to describe set C. (The basket represents set C.) C = {comb, brush, toothbrush, paste, soap, razor, cloth}

C = {x|x is an object used in your morning cleanup} EXAMPLE 2 Use set-builder notation to describe set C. C = {x|x is an object used in your morning cleanup}

Set-builder: Listing: {x|x is a ball used in a major sport} {baseball, volleyball, basketball, soccer ball, football}

EXAMPLE 3 Let A = {1, 3, 5, 7, 9} and B = {x|x is an odd number less than 10}. Symbolize a relation between A and B. B = {1, 3, 5, 7, 9} A = B

If set A contains set B, then set B is a subset of set A and we write B  A. Each element of B must also be an element of set A. Every set is a subset of itself: A  A.

The empty set, or null set, denoted by { } or Ø, is the set that contains no elements. The empty set is a subset of every set. Therefore, Ø  A.

EXAMPLE 4 If P = {2, 7, 12, 17} and Q = {2, 12}, name three subsets of P. Q  P, P  P, and Ø  P.

A is a proper subset of B if it is a subset of B and A ≠ B A is a proper subset of B if it is a subset of B and A ≠ B. A proper subset is denoted by the symbol .

Notation Subset: B  A Proper subset: B  A Equal sets: B = A Empty Set: Ø or { } Member (Element) of: 

Equal sets are sets with the same elements Equal sets are sets with the same elements. Equivalent sets are sets that are in one-to-one correspondence.

One-to-one correspondence – two sets having the same number of elements, equinumerous

The universal set, denoted by U, is the set of all elements under discussion for a given problem. Venn diagrams are often used to illustrate sets and their relationships.

A Venn diagram is a diagram in which mathematical sets are represented by overlapping circles within a boundary representing the universal set, so that all possible combinations of the relevant properties are represented by distinct areas of the diagram.

Homework p. 5

►A. Exercises Write the following sets with both listing statements and set-builder notation. 1. States that border your state. Listing: {Georgia, North Carolina} Set-builder notation: {x|x is a state that borders South Carolina}

►A. Exercises Tell whether the following pairs of sets are (1) equivalent sets, (2) equal sets, or (3) neither. Give the most specific answer possible. 7. K = {5,7,8} M = {1, 9, 2, 7} (3) Neither

►A. Exercises Tell whether the following pairs of sets are (1) equivalent sets, (2) equal sets, or (3) neither. Give the most specific answer possible. 9. L = {man, son, brother} N = {woman, daughter, sister} (1) Equivalent

►A. Exercises Use the proper notation to describe these statements. 11. Set A is a subset of set L. 1. {A}  {L} 2. {A}  {L} 3. {A}  {L} 4. A  L 5. A  L

►A. Exercises Use the proper notation to describe these statements. 13. The empty set is a subset of the universal set. 1. {0}  U 2. {Ø}  U 3. { }  U 4. Ø  U

►A. Exercises Use the proper notation to describe these statements. 15. Set K is equal to set F. 1. K = F 2. {K} = {F} 3. {K}  {F}

►A. Exercises Use the proper notation to describe these statements. 17. Set N is not a subset of L. N  L

►A. Exercises Use the proper notation to describe these statements. 19. The set with elements k, l, and m is not a subset of the set consisting of elements k, l, and n. {k, l, m}  {k, l, n}

►B. Exercises Draw Venn diagrams to illustrate the following sets. Give the universal set in each case. 23. N = {1,2,3,4,5,6} P = {1,3,5} U N 2 4 6 P 1 3 5