1. Yet another important mathematical concept
Functions
Yet another important mathematical concept

2. Functions
Most basic representation: Arrow Diagrams B A 𝑓

3. Functions Most basic representation: Arrow Diagrams
A is called the domain and B is called the co-domain and we say that “f is such that it maps elements from A to B” (𝑓:𝐴↦𝐵) B A 𝑓
Linger on this

4. Functions Most basic representation: Arrow Diagrams
A is called the domain and B is called the co-domain and we say that “f is such that it maps elements from A to B” (𝑓:𝐴↦𝐵) B A 𝑓
\mapsto in LaTeX

5. Example 1
Yes No
Is this a function? A B 𝑓 1 2 4 3

6. Example 1 Yes No Is this a function? 𝑓 A B Domain: {0, 1, 2, 3, 4}
Co-domain: {0, 1, 4}
Formula (that we came up with): 𝑓 𝑥 = 𝑥 2 % 5 (where % is the MOD operation) A B 𝑓 1 2 4 3

7. Example 2
Yes No
Is this a function? A B 1 2 4 3

8. Example 2 Yes No Is this a function? 3 A B
Every element of the domain should map to some co-domain element! A B 1 2 4 3

9. Example 3
Yes No
Is this a function?
𝑓:ℕ↦ℕ, and 𝑓 𝑥 = 𝑥 2

10. Example 3 Yes No Is this a function? 𝑓:ℕ↦ℕ, and 𝑓 𝑥 = 𝑥 2
For any odd selection of 𝑥∈ℕ, there is no 𝑥 2 ∈ ℕ!

11. Arrow Diagrams: Pros and Cons
Good for introducing functions as mappings
No information on other function properties (continuous, integrable, monotonic, …)
Help us decouple functions from other mappings in 𝐴 ×𝐵
Only possible for functions with small, finite domains and co-domains
Clarify what the domain and what the co-domain is ….
Good for teaching us what onto and 1-1 means …

12. Arrow Diagrams: Pros and Cons
Good for introducing functions as mappings
No information on other function properties (continuous, integrable, monotonic, …)
Help us decouple functions from other mappings in 𝐴 ×𝐵
Only possible for functions with small, finite domains and co-domains
Clarify what the domain and what the co-domain is …
Good for teaching us what onto and 1-1 means
Those are important cons….

13. Graphs
We can also represent functions by their graphs…

14. (“vertical line test”)
Graphs (pros,cons)
Which has its owns pros and cons
Pros
Cons
Easy to determine 1-1 functions (“horizontal line test”)
Cannot compute the derivative at any point, only draw it
Easy to determine non-functions
(“vertical line test”)
Cannot compute the value of the function at any point
Can determine monotonicity and continuity easily
Cannot compute the indefinite integral of the function
Can easily determine if the function intersects the axes
Cannot determine where it intersects the axes …

15. Short quiz

16. Short quiz
Yes No
Are the following valid functions?

17. Short quiz
Yes No
Are the following valid functions?

18. Short quiz
Yes No
Are the following valid functions?
Log function

19. Short quiz
Yes No
Are the following valid functions?

20. Short quiz Yes No Fails the “vertical line” test
Are the following valid functions?
Fails the “vertical line” test

21. Short quiz
Yes No
Are the following valid functions? B A 𝑓

22. Short quiz Yes No Are the following valid functions? 𝑓 B A
It just happens to
not be “1-1”
(term we’ll explain momentarily)!

23. Function formulae
Yes No
Are the following valid functions? B A 𝑓

24. Function formulae Yes No
Are the following valid functions? B A 𝑓
Fails the “vertical line” test (2 different `y’s mapped to by the same `x’)

25. Surjective functions
A function 𝑓:𝑋↦𝑌 is called surjective (or onto) iff
∀𝑦∈𝑌, ∃𝑥∈𝑋:𝑓 𝑥 =𝑦

26. Surjective functions
A function 𝑓:𝑋↦𝑌 is called surjective (or onto) iff
∀𝑦∈𝑌, ∃𝑥∈𝑋:𝑓 𝑥 =𝑦
Intuitively: 𝑓 ′ 𝑠 co-domain is “full” with pointers

27. Surjective functions
A function 𝑓:𝑋↦𝑌 is called surjective (or onto) iff
∀𝑦∈𝑌, ∃𝑥∈𝑋:𝑓 𝑥 =𝑦
Intuitively: 𝑓 ′ 𝑠 co-domain is “full” with pointers
Mnemonic rule to remember the name “surjective”: in French, “sur” means “above”, “on”, or “onto”.

28. Surjective functions
A function 𝑓:𝑋↦𝑌 is called surjective (or onto) iff
∀𝑦∈𝑌, ∃𝑥∈𝑋 [𝑓 𝑥 =𝑦]
Intuitively: 𝑓 ′ 𝑠 co-domain is “full” with pointers
Mnemonic rule to remember the name “surjective”: in French, “sur” means “above”, “on”, or “onto”.
“sur”

29. Quiz on surjectivity
Yes No
Something Else
Is 𝑓 𝑥 = 𝑥 2 surjective?

30. Quiz on surjectivity Yes No Is 𝑓 𝑥 = 𝑥 2 surjective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 surjective?
We have not provided you with either the domain or the co-domain!

31. Quiz on surjectivity Yes No Is 𝑓 𝑥 = 𝑥 2 surjective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 surjective?
We have not provided you with either the domain or the co-domain!
𝐷=ℝ, 𝐶=ℝ: Non-Surjective! (e.g -1 is not mapped to)

32. Quiz on surjectivity Yes No Is 𝑓 𝑥 = 𝑥 2 surjective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 surjective?
We have not provided you with either the domain or the co-domain!
𝐷=ℝ, 𝐶=ℝ: Non-Surjective! (e.g -1 is not mapped to)
𝐷=ℝ,𝐶= ℝ ≥0 : Surjective!

33. Quiz on surjectivity Yes No Is 𝑓 𝑥 = 𝑥 2 surjective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 surjective?
We have not provided you with either the domain or the co-domain!
𝐷=ℝ, 𝐶=ℝ: Non-Surjective! (e.g -1 is not mapped to)
𝐷=ℝ,𝐶= ℝ ≥0 : Surjective!
𝐷=ℕ,𝐶=? such that it is surjective?

34. Quiz on surjectivity Yes No Is 𝑓 𝑥 = 𝑥 2 surjective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 surjective?
We have not provided you with either the domain or the co-domain!
𝐷=ℝ, 𝐶=ℝ: Non-Surjective! (e.g -1 is not mapped to)
𝐷=ℝ,𝐶= ℝ ≥0 : Surjective!
𝐷=ℕ,𝐶=𝑆𝑄𝑈𝐴𝑅𝐸𝑆: Surjective!

35. Injective functions A function 𝑓:𝑋↦𝑌 is called injective (or 1-1) iff
(∀ 𝑥 1 , 𝑥 2 ∈𝑋) [𝑓 𝑥 1 =𝑓 𝑥 2 ⇒ 𝑥 1 = 𝑥 2 ]
Intuitively: Every element of the co-domain is mapped to by at most one element of the domain.

36. Injective functions A function 𝑓:𝑋↦𝑌 is called injective (or 1-1) iff
(∀ 𝑥 1 , 𝑥 2 ∈𝑋) [𝑓 𝑥 1 =𝑓 𝑥 2 ⇒ 𝑥 1 = 𝑥 2 ]
Intuitively: Every element of the co-domain is mapped to by at most one element of the domain.
Why “at most one” and not exactly one?

37. Injective functions A function 𝑓:𝑋↦𝑌 is called injective (or 1-1) iff
(∀ 𝑥 1 , 𝑥 2 ∈𝑋) [𝑓 𝑥 1 =𝑓 𝑥 2 ⇒ 𝑥 1 = 𝑥 2 ]
Intuitively: Every element of the co-domain is mapped to by at most one element of the domain.
Why at most one and not exactly one?
Because 1-1 but not onto functions are
possible! X Y 𝑓

38. Quiz on injectivity
Yes No
Something Else
Is 𝑓 𝑥 = 𝑥 2 injective?

39. Quiz on injectivity Yes No Is 𝑓 𝑥 = 𝑥 2 injective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 injective?
We need domains and codomains!

40. Quiz on injectivity Yes No Is 𝑓 𝑥 = 𝑥 2 injective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 injective?
We need domains and codomains!
𝐷=ℝ, 𝐶=ℝ: Non-Injective! (e.g −3 2 = 3 2 =9)

41. Quiz on injectivity Yes No Is 𝑓 𝑥 = 𝑥 2 injective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 injective?
We need domains and codomains!
𝐷=ℝ, 𝐶=ℝ: Non-Injective! (e.g −3 2 = 3 2 =9)
𝐷=ℝ,𝐶= ℝ ≥0 : Non−Injective! (same example works)

42. Quiz on injectivity Yes No Is 𝑓 𝑥 = 𝑥 2 injective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 injective?
We need domains and codomains!
𝐷=ℝ, 𝐶=ℝ: Non-Injective! (e.g −3 2 = 3 2 =9)
𝐷=ℝ,𝐶= ℝ ≥0 : Non−Injective! (same example works)
𝐷=ℤ,𝐶=𝑆𝑄𝑈𝐴𝑅𝐸𝑆: Non−Injective! (same example works)

43. Quiz on injectivity Yes No Is 𝑓 𝑥 = 𝑥 2 injective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 injective?
We need domains and codomains!
𝐷=ℝ, 𝐶=ℝ: Non-Injective! (e.g −3 2 = 3 2 =9)
𝐷=ℝ,𝐶= ℝ ≥0 : Non−Injective! (same example works)
𝐷=ℤ,𝐶=𝑆𝑄𝑈𝐴𝑅𝐸𝑆: Non−Injective! (same example works)
Can this function ever be injective?

44. Quiz on injectivity Yes No Is 𝑓 𝑥 = 𝑥 2 injective?
Something Else
Is 𝑓 𝑥 = 𝑥 2 injective?
We need domains and codomains!
𝐷=ℝ, 𝐶=ℝ: Non-Injective! (e.g −3 2 = 3 2 =9)
𝐷=ℝ,𝐶= ℝ ≥0 : Non−Injective! (same example works)
𝐷=ℤ,𝐶=𝑆𝑄𝑈𝐴𝑅𝐸𝑆: Non−Injective! (same example works)
Can this function ever be injective?
Yes. Pick 𝐷=ℕ,𝐶=𝑆𝑄𝑈𝐴𝑅𝐸𝑆

45. Making functions onto or 1-1
To make a function onto, we need to make the
co-domain smaller.
To make a function 1-1 , we need to make the
domain smaller.

46. Bijective functions
A function 𝑓:𝑋↦𝑌 is called bijective (or a bijection, or a 1- 1 correspondence) iff it is both surjective and injective.
We will try to avoid using the term “1-1 correspondence” (your book uses it) since it can confuse us with the notion of an injective (or 1-1) function.

47. Quiz on bijections

48. Quiz on bijections Yes No Are the following functions bijections?
(In all examples, 𝐶=ℝ)

49. Quiz on bijections Yes No Are the following functions bijections?
𝑓 𝑥 = 𝑥 , 𝑥∈ℝ
(In all examples, 𝐶=ℝ)

50. Quiz on bijections Yes No Are the following functions bijections?
(In all examples, 𝐶=ℝ)
Non-injective!

51. Quiz on bijections Yes No Are the following functions bijections?
𝑓 𝑥 = 𝑥 , 𝑥∈ℝ
𝑓 𝑥 =𝑎⋅𝑥+𝑏, 𝑎,𝑥,𝑏∈ℝ
(In all examples, 𝐶=ℝ)

52. Quiz on bijections Yes No
Are the following functions bijections?
𝑓 𝑥 = 𝑥 , 𝑥∈ℝ No
𝑓 𝑥 =𝑎⋅𝑥+𝑏, 𝑎,𝑥,𝑏∈ℝ No
(In all examples, 𝐶=ℝ)
!!!!
For a = 0, the graph of the function fails the “horizontal line test”!

53. Quiz on bijections Yes No Are the following functions bijections?
𝑔 𝑥 = 𝑎⋅𝑥 2 , 𝑎,𝑥∈ ℝ, 𝑎>0
(In all examples, 𝐶=ℝ)

54. Quiz on bijections Yes No Are the following functions bijections?
(In all examples, 𝐶=ℝ)
Non-injective!

55. Quiz on bijections Yes No Are the following functions bijections?
ℎ 𝑛 =4𝑛 −1, 𝑛∈ℤ
(In all examples, 𝐶=ℝ)

56. Quiz on bijections Yes No Are the following functions bijections?
(In all examples, 𝐶=ℝ)
Non-surjective! Set ℎ 𝑛 =𝑦 and solve for 𝑛:
4𝑛 – 1 = 𝑦⇒𝑛= 𝑦+1 4
There are infinitely many choices of 𝑦 for which 𝑛∉ℤ!

57. Quiz on bijections Yes No Are the following functions bijections?
ℎ 𝑥 =4𝑥 −1, 𝑥∈ℝ
(In all examples, 𝐶=ℝ)

58. Quiz on bijections Yes No Are the following functions bijections?
(In all examples, 𝐶=ℝ)
Surjective and injective! Surjective, since, if we set ℎ 𝑛 =𝑦 and solve for 𝑛:
4𝑛 – 1 = 𝑦⇒𝑛= 𝑦+1 4
For every real 𝑦, there’s always a real solution 𝑛. Injective, since it’s of the form of (2) with 𝑎≠0.

59. Some interesting functions
We will examine a pair of interesting functions from ℝ to ℤ.
Given a real number 𝑟, the floor of 𝑟, denoted 𝑟 is the largest integer 𝑛:𝑛≤𝑟.

60. Floor and ceiling
We will examine a pair of interesting functions from ℝ to ℤ.
Given a real number 𝑟, the floor of 𝑟, denoted 𝑟 is the largest integer 𝑛:𝑛≤𝑟.
Given a real number 𝑟, the ceiling of 𝑟, denoted ⌈𝑟⌉ is the smallest integer 𝑛:𝑛≥𝑟.

61. Floor and ceiling
We will examine a pair of interesting functions from ℝ to ℤ.
Given a real number 𝑟, the floor of 𝑟, denoted 𝑟 is the largest integer 𝑛:𝑛≤𝑟.
Given a real number 𝑟, the ceiling of 𝑟, denoted ⌈𝑟⌉ is the smallest integer 𝑛:𝑛≥𝑟.
⌊ 𝑟 1 ⌋
⌈ 𝑟 1 ⌉
⌊ 𝑟 2 ⌋
⌈ 𝑟 2 ⌉ -4 -3 -2 -1 1 2 3 4 5 6 7
𝑟 1
𝑟 2

62. Examples
1.3 =1

63. Examples
1.3 =1
10.3 =10

64. Examples
1.3 =1
10.3 =10
0.8 =1

65. Examples
1.3 =1
10.3 =10
0.8 =1
…1 =1

66. Examples
1.3 =1
10.3 =10
0.8 =1
…1 =1
−5.6 =−5
At this point, might be worth mentioning that those functions are not injective!

67. Examples From these two we can deduce… 1.3 =1 10.3 =10 0.8 =1
…1 =1
−5.6 =−5
From these two we can deduce…
At this point, might be worth mentioning that those functions are not injective!

68. Examples From these two we can deduce… that floor is non-injective!
1.3 =1
10.3 =10
0.8 =1
…1 =1
−5.6 =−6
From these two we can deduce… that floor is non-injective!
(same for ceiling, of course)
At this point, might be worth mentioning that those functions are not injective!

69. Vote!
3 = 3 2 4
3.5

70. Vote!
3 =3
𝜋 = 3 2 4
3.5 3 𝜋 4
3.5

71. Vote! 3 2 4
3.5
3 =3
𝜋 = 4
2 500 × = 3 𝜋 4
3.5
2 500 ×500
2 500 × × 500 2
2 499 × 500 2
Something else

72. Vote! 3 2 4
3.5
3 =3
𝜋 = 4
2 500 × = × 500 2
∀𝑛∈ℤ, 𝑛 = 𝑛 =𝑛 3 𝜋 4
3.5
2 500 ×500
2 500 × × 500 2
2 499 × 500 2
Something else T 𝐹

73. Vote! 3 =3 𝜋 = 4 ∀𝑛∈ℤ, 𝑛 = 𝑛 =𝑛 T ∀𝑛∈ℤ, 𝑛 = 𝑛 =𝑛 3 2 4 3.5 3 𝜋 4 3.5 T
3 =3
𝜋 = 4
2 500 × = × 500 2
∀𝑛∈ℤ, 𝑛 = 𝑛 =𝑛 T
∀𝑛∈ℤ, 𝑛 = 𝑛 =𝑛 3 𝜋 4
3.5
2 500 ×500
2 500 × × 500 2
2 499 × 500 2
Something else T 𝐹 T 𝐹

74. Vote! 3 =3 𝜋 = 4 ∀𝑛∈ℤ, 𝑛 = 𝑛 =𝑛 T ∀𝑛∈ℤ, 𝑛 = 𝑛 =𝑛 T 3 2 4 3.5 3 𝜋 4 3.5
3 =3
𝜋 = 4
2 500 × = × 500 2
∀𝑛∈ℤ, 𝑛 = 𝑛 =𝑛 T
∀𝑛∈ℤ, 𝑛 = 𝑛 =𝑛 T 3 𝜋 4
3.5
2 500 ×500
2 500 × × 500 2
2 499 × 500 2
Something else T 𝐹 T 𝐹

75. Relations

76. Functions ultra-formally
Consider the function that we talked about before: A B 𝑓 1 2 4 3

77. Functions ultra-formally
Consider the function that we talked about before:
Can view it as the set of ordered pairs { 0,0 , 1,1 , 2,4 , 3,4 , (4, 1)} A B 𝑓 1 2 4 3

78. Functions ultra-formally
Let 𝐴, 𝐵 be sets. A function 𝑓:𝐴↦𝐵 is any subset 𝑆=𝐴 × 𝐵 where
∀𝑎∈𝐴 [ ∃𝑏∈𝐵 𝑎, 𝑏 ∈𝑆 ] (every element of the domain maps somewhere)
∀𝑎∈𝐴 [ ∃ 𝑏 1 ∈𝐵 𝑎, 𝑏 1 ∈𝑆 ∧∀ 𝑏 2 ≠ 𝑏 1 𝑎, 𝑏 2 ∉𝑆 ] (every element of the domain maps to exactly one element of the co-domain)

79. Functions ultra-formally
Let 𝐴, 𝐵 be sets. A function 𝑓:𝐴↦𝐵 is any subset 𝑆=𝐴 × 𝐵 where
∀𝑎∈𝐴 [ ∃𝑏∈𝐵 𝑎, 𝑏 ∈𝑆 ] (every element of the domain maps somewhere)
∀𝑎∈𝐴 [ ∃ 𝑏 1 ∈𝐵 𝑎, 𝑏 1 ∈𝑆 ∧∀ 𝑏 2 ≠ 𝑏 1 𝑎, 𝑏 2 ∉𝑆 ] (every element of the domain maps to exactly one element of the co-domain)
We generalize the notion of a function to that of a relation!
Any subset of 𝐴 × 𝐵 that doesn’t abide by (1) or (2) we call a relation from A to B.

80. Examples (<, ℝ×ℝ) {…, −1.5, −1.2 , −1.4,−1.2 , 2 , 3 , 2 , 5 ,…}
{…, −1.5, −1.2 , −1.4,−1.2 , 2 , 3 , 2 , 5 ,…}
Verbally tell them that is is “less than” on R^2

81. Examples (<, ℝ×ℝ) ≤,ℝ×ℝ
{…, −1.5, −1.2 , −1.4,−1.2 , 2 , 3 , 2 , 5 ,…}
≤,ℝ×ℝ
{…, 2, 2 , 2, 2.1 , 2 , 3 , 2 , 5 ,…}

82. Examples (<, ℝ×ℝ) ≤,ℝ×ℝ 𝐵𝑖𝑙𝑙, ℝ×ℕ
{…, −1.5, −1.2 , −1.4,−1.2 , 2 , 3 , 2 , 5 ,…}
≤,ℝ×ℝ
{…, 2, 2 , 2, 2.1 , 2 , 3 , 2 , 5 ,…}
𝐵𝑖𝑙𝑙, ℝ×ℕ
{ 𝑟, 𝑛 | 𝑛 appears in the decimal expansion of 𝑟 }
E.g: {…, 𝜋, 1 , 𝑒, 7 , , 4 , …}
We would formally say that all of the above are elements of the relation “Bill”
Verbally state the names of the other relations.

83. Reflexivity
A relation X⊆𝐴 × 𝐴 is reflexive if
∀𝑎∈𝐴 [ 𝑎, 𝑎 ∈𝑋 ]

84. Reflexivity A relation X⊆𝐴 × 𝐴 is reflexive if ∀𝑎∈𝐴 𝑎, 𝑎 ∈𝑋 Examples:
∀𝑎∈𝐴 𝑎, 𝑎 ∈𝑋
Examples:
(≤,ℕ×ℕ) is reflexive, since ∀𝑛∈ℕ 𝑛≤𝑛
(<,ℕ×ℕ) is not reflexive, since ∼ ∀𝑛∈ℕ 𝑛<𝑛 (in fact, there is no such 𝑛)
(𝐽𝑎𝑠𝑜𝑛,ℕ×ℕ) defined as { 𝑥, 𝑦 | 𝑥+𝑦≥100} is not reflexive (e.g 10, 10 ∉𝐽𝑎𝑠𝑜𝑛))
Say verbally that one needs only one counter-example (a forall statement can be proven negative with just one witness)

85. Symmetry A relation X⊆𝐴 × 𝐴 is symmetric if
∀ 𝑎 1 , 𝑎 2 ∈𝐴 𝑎 1 , 𝑎 2 ∈𝑋 ⇒ 𝑎 2 , 𝑎 1 ∈𝑋)

86. Symmetry A relation X⊆𝐴 × 𝐴 is symmetric if
∀ 𝑎 1 , 𝑎 2 ∈𝐴 𝑎 1 , 𝑎 2 ∈𝑋 ⇒ 𝑎 2 , 𝑎 1 ∈𝑋)
Examples:
(≤,ℕ×ℕ) is not symmetric since 4≤5 but ∼(5≤4)
(<,ℕ×ℕ) is not symmetric (see above)
(𝐽𝑎𝑠𝑜𝑛,ℕ×ℕ) defined as { 𝑥, 𝑦 | 𝑥+𝑦≥100} is symmetric since
𝑥+𝑦≤100 ⇒(𝑦+𝑥≤100)

87. ∀ 𝑎 1 , 𝑎 2 , 𝑎 3 ∈𝐴 𝑎 1 , 𝑎 2 ∈𝑋 ∧( 𝑎 2 , 𝑎 3 ∈𝑋)⇒ 𝑎 1 , 𝑎 3 ∈𝑋)
Transitivity
A relation X⊆𝐴 × 𝐴 is transitive if
∀ 𝑎 1 , 𝑎 2 , 𝑎 3 ∈𝐴 𝑎 1 , 𝑎 2 ∈𝑋 ∧( 𝑎 2 , 𝑎 3 ∈𝑋)⇒ 𝑎 1 , 𝑎 3 ∈𝑋)

88. ∀ 𝑎 1 , 𝑎 2 , 𝑎 3 ∈𝐴 𝑎 1 , 𝑎 2 ∈𝑋 ∧( 𝑎 2 , 𝑎 3 ∈𝑋)⇒ 𝑎 1 , 𝑎 3 ∈𝑋)
Transitivity
A relation X⊆𝐴 × 𝐴 is transitive if
∀ 𝑎 1 , 𝑎 2 , 𝑎 3 ∈𝐴 𝑎 1 , 𝑎 2 ∈𝑋 ∧( 𝑎 2 , 𝑎 3 ∈𝑋)⇒ 𝑎 1 , 𝑎 3 ∈𝑋)
(≤,ℕ×ℕ) is transitive since 𝑥≤𝑦 ∧ 𝑦≤𝑧 ⇒ 𝑥≤𝑧
(<,ℕ×ℕ) is transitive (see above)
(𝐽𝑎𝑠𝑜𝑛,ℕ×ℕ) defined as { 𝑥, 𝑦 | 𝑥+𝑦≥100} ???

89. Transitivity A relation X⊆𝐴 × 𝐴 is transitive if
∀ 𝑎 1 , 𝑎 2 , 𝑎 3 ∈𝐴 𝑎 1 , 𝑎 2 ∈𝑋 ∧( 𝑎 2 , 𝑎 3 ∈𝑋)⇒ 𝑎 1 , 𝑎 3 ∈𝑋)
(≤,ℕ×ℕ) is transitive since 𝑥≤𝑦 ∧ 𝑦≤𝑧 ⇒ 𝑥≤𝑧
(<,ℕ×ℕ) is transitive (see above)
(𝐽𝑎𝑠𝑜𝑛,ℕ×ℕ) defined as { 𝑥, 𝑦 | 𝑥+𝑦≥100} is not transitive since (counter-example):
1, 100 ∈𝐽𝑎𝑠𝑜𝑛∧ 100, 5 ∈𝐽𝑎𝑠𝑜𝑛 but 1, 5 ∉𝐽𝑎𝑠𝑜𝑛