1. Properties and Relations
1

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?
7

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?
8

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?
9

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?
10

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?
11

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?
12

13. Reflexive Property
• For any number a, a = a.
• For what sets of numbers is this property
valid?
13

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?
14

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?
15

16. Identity Relation
• Because equality exhibits the reflexive,
symmetric, and transitive properties, it is
called an equivalence relation.
16

17. Equivalence Relation
• Any relation that has the reflexive,
symmetric, and transitive properties is an
equivalence relation.
17

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
18

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
22

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
24

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
28

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
30

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
31

32. Now It’s Your Turn
• Find some sets and relations that
represent equivalence relations.
32