The document discusses properties of relations and equivalence relations. It examines whether relations like "has the same color eyes as" or "is similar to" satisfy the properties of reflexivity, symmetry, and transitivity needed to be an equivalence relation for different sets. Some key examples determined to be equivalence relations are relations of congruence on triangles and differences being divisible by 2 for integers. The document prompts the reader to explore additional examples of equivalence relations.

1. Properties and Relations
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2. Closure Property for Addition
• For any two numbers a and b, a+b is
unique and a+b is contained in the set of
numbers.
• For what sets of numbers is this property
valid?
• Find a set of numbers for which this
property is not valid.
2

3. Closure Property for
Multiplication
• For any two numbers a and b, ab is unique
and ab is contained in the set of numbers.
• For what sets of numbers is this property
valid?
• Find a set of numbers for which this
property is not valid.
3

4. Commutative Property for
Addition
• For any two numbers a and b, a+b = b+a.
• For what sets of numbers is this property
valid?
4

5. Commutative Property for
Multiplication
• For any two numbers a and b, ab = ba.
• For what sets of numbers is this property
valid?
5

6. Associative Property for
Addition
• For any three numbers a, b, and c,
a + (b + c) = (a + b) + c.
• For what sets of numbers is this property
valid?
6

7. Associative Property for
Multiplication
• For any three numbers a, b, and c,
a(bc) = (ab)c.
• For what sets of numbers is this property
valid?
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8. Distributive Property
• For any three numbers a, b, and c,
a(b + c) = ab + ac.
• For what sets of numbers is this property
valid?
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9. Additive Identity Property
• For any number a, a + 0 = 0 + a = a. Zero
is called the additive identity.
• For what sets of numbers is this property
valid?
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10. Additive Inverse Property
• For any number a, there exists a number
–a (read the opposite of a) such that
a + –a = 0.
• For what sets of numbers is this property
valid?
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11. Multiplicative Identity Property
• For any number a, a(1) = 1(a) = a. One is
called the multiplicative identity.
• For what sets of numbers is this property
valid?
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12. Multiplicative Inverse Property
• For any number a, there exists a number
1/a (read the reciprocal of a) such that
a(1/a) = 1.
• For what sets of numbers is this property
valid?
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13. Reflexive Property
• For any number a, a = a.
• For what sets of numbers is this property
valid?
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14. Symmetric Property
• For any two numbers a and b, if a = b then
b = a.
• For what sets of numbers is this property
valid?
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15. Transitive Property
• For any three numbers a, b, and c, if a = b
and b = c, then a = c.
• For what sets of numbers is this property
valid?
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16. Identity Relation
• Because equality exhibits the reflexive,
symmetric, and transitive properties, it is
called an equivalence relation.
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17. Equivalence Relation
• Any relation that has the reflexive,
symmetric, and transitive properties is an
equivalence relation.
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18. Let’s Explore Relations
• Determine if the relation “has the same
color of eyes as” on the set of “people who
live in the USA” is an equivalence relation.
• Reflexive? Yes
• Symmetric? Yes
• Transitive? Yes
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19. Let’s Explore Relations
• Determine if the relation “has the same
color of eyes as” on the set of “people who
live in the USA” is an equivalence relation.
• Reflexive? Yes
• Symmetric? Yes
• Transitive? Yes
19Therefore, this IS an equivalence relation.

20. Let’s Explore Relations
• Determine if the relation “is a classmate
of” on the set of “students currently
enrolled in MATH 517” is an equivalence
relation.
• Reflexive? No
• Symmetric? Yes
• Transitive? Yes
20

21. Let’s Explore Relations
• Determine if the relation “is a classmate
of” on the set of “students currently
enrolled in MATH 517” is an equivalence
relation.
• Reflexive? No
• Symmetric? Yes
• Transitive? Yes
21Therefore, this is NOT an equivalence relation.

22. Let’s Explore Relations
• Determine if the relation “is similar to” on
the set of “quadrilaterals” is an
equivalence relation.
• Reflexive? Yes
• Symmetric? Yes
• Transitive? Yes
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23. Let’s Explore Relations
• Determine if the relation “is similar to” on
the set of “quadrilaterals” is an
equivalence relation.
• Reflexive? Yes
• Symmetric? Yes
• Transitive? Yes
23Therefore, this IS an equivalence relation.

24. Let’s Explore Relations
• Determine if the relation “has the same or
more volume than” on the set of “3-D
figures” is an equivalence relation.
• Reflexive? Yes
• Symmetric? No
• Transitive? Yes
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25. Let’s Explore Relations
• Determine if the relation “has the same or
more volume than” on the set of “3-D
figures” is an equivalence relation.
• Reflexive? Yes
• Symmetric? No
• Transitive? Yes
25Therefore, this is NOT an equivalence relation.

26. Let’s Explore Relations
• Determine if the relation “has the same
number of factors as” on the set of “prime
numbers” is an equivalence relation.
• Reflexive? Yes
• Symmetric? Yes
• Transitive? Yes
26

27. Let’s Explore Relations
• Determine if the relation “has the same
number of factors as” on the set of “prime
numbers” is an equivalence relation.
• Reflexive? Yes
• Symmetric? Yes
• Transitive? Yes
27Therefore, this IS an equivalence relation.

28. Let’s Explore Relations
• Determine if the relation “have a common
factor greater than 1” on the set {2, 3, 4,
…} is an equivalence relation.
• Reflexive? Yes
• Symmetric? Yes
• Transitive? No
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29. Let’s Explore Relations
• Determine if the relation “have a common
factor greater than 1” on the set of
“composite numbers” is an equivalence
relation.
• Reflexive? Yes
• Symmetric? Yes
• Transitive? No
29Therefore, this is NOT an equivalence relation.

30. Let’s Explore Relations
Set Relation Reflexive? Symmetric? Transitive? Equivalence
Relation?
People Is a sister of
People Is a sibling
of
Triangles Are
congruent
Lines in a
plane
Have at
least one
point in
common
Integers Difference
is divisible
by 2
Natural
numbers
Divisible by
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31. Let’s Explore Relations
Set Relation Reflexive? Symmetric? Transitive? Equivalence
Relation?
People Is a sister of No No Yes No
People Is a sibling
of
No Yes Yes No
Triangles Are
congruent
Yes Yes Yes Yes
Lines in a
plane
Have at
least one
point in
common
Yes Yes No No
Integers Difference
is divisible
by 2
Yes Yes Yes Yes
Natural
numbers
Divisible by Yes No Yes No
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32. Now It’s Your Turn
• Find some sets and relations that
represent equivalence relations.
32