1. Math
behind
software

2. How are math &
software related?

3. How are math &
software related?
Binary
System

4. How are math &
software related?
Binary
System
Algorithms

5. How are math &
software related?
Binary
System
Algorithms
CAP
theorem

6. How are math &
software related?
Binary
System
Algorithms
CAP
theorem
Monads

7. What will be the value of B

8. What will be the value of B
True

9. What will be the value of B
True False

10. True

11. Why calling allMatch on
empty Stream always
return true?

12. Why calling allMatch on
empty Stream always
return true?
...And what does it have
to do with math?

13. Vacuous truth

14. Vacuous truth

15. Vacuous truth
Is a concept from logic used to describe all
statements that are evaluated as true because
their base condition (Antecedent) cannot be
fulfilled

16. Vacuous truth
Is a concept from logic used to describe all
statements that are evaluated as true because
their base condition (Antecedent) cannot be
fulfilled
Or in a more mathematical syntax

17. S: P ⇒Q

18. S: P ⇒Q
Then, if P is false, S is logically true and considered
vacuously true.

19. S: P ⇒Q
Then, if P is false, S is logically true and considered
vacuously true.
It can also be defined with quantificator - for all ∀:

20. S: P ⇒Q
Then, if P is false, S is logically true and considered
vacuously true.
It can also be defined with quantificator - for all ∀:
∀x ∈B : P(x) ⇒Q(x)

21. S: P ⇒Q
Then, if P is false, S is logically true and considered
vacuously true.
It can also be defined with quantificator - for all ∀:
∀x ∈B : P(x) ⇒Q(x)
Then, if P(x) is false for all x, our statement is also
considered vacuously true.

22. Example
If there are any developers in the
room next door, then they all work
with Haskell.

23. Example
If there are any developers in the
room next door, then they all work
with Haskell.
the base condition is: If there are any
developers in the room next door

24. Example
If there are any developers in the
room next door, then they all work
with Haskell.
the base condition is: If there are any
developers in the room next door
if the room's empty, the statement is
evaluated as vacuously true

25. Haskell devs & mixed syntax

26. Haskell devs & mixed syntax
P(x) is our explicit base condition - there are any
developers in the room

27. Haskell devs & mixed syntax
P(x) is our explicit base condition - there are any
developers in the room
Q(x) is our 'then' clause - they all work in Haskell

28. Haskell devs & mixed syntax
P(x) is our explicit base condition - there are any
developers in the room
Q(x) is our 'then' clause - they all work in Haskell
B is our set - all the people in the room

29. Haskell devs & mixed syntax
P(x) is our explicit base condition - there are any
developers in the room
Q(x) is our 'then' clause - they all work in Haskell
B is our set - all the people in the room
We can write the example in this form:
For all the people in the room, if a person is a developer,
then they work with Haskell.

30. How exactly do Stream
and allMatch fit into this?

31. How exactly do Stream
and allMatch fit into this?
∀x ∈B : P(x) ⇒Q(x)

33. Similar behavior in other languages

34. Similar behavior in other languages
In Scala: forall, present in many collections

35. Similar behavior in other languages
In Scala: forall, present in many collections
In JavaScript: every from array prototype

36. Similar behavior in other languages
In Scala: forall, present in many collections
In JavaScript: every from array prototype
In Python & Kotlin: all - which tests if all
elements of an array are true, will also return
true for empty arrays

37. Vacuous truth daily

38. Vacuous truth daily
when hell freezes

39. Vacuous truth daily
when hell freezes
when pigs fly

40. Vacuous truth daily
when hell freezes
when pigs fly
when shrimps learn to whistle

41. Vacuous truth daily
when hell freezes
when pigs fly
when shrimps learn to whistle
when the sun starts rising in the west

42. Vacuous truth daily
when hell freezes
when pigs fly
when shrimps learn to whistle
when the sun starts rising in the west
...or any other absurdity based on an obviously
false or impossible condition

43. Vacuous truth - recap:

44. Vacuous truth - recap:
is a concept from the mathematics field of
logic

45. Vacuous truth - recap:
is a concept from the mathematics field of
logic
represents logical statements with a false base
condition

46. Vacuous truth - recap:
is a concept from the mathematics field of
logic
represents logical statements with a false base
condition
is the reason why allMatch and their
counterparts will return true for empty
sequences

47. Vacuous truth - recap:
is a concept from the mathematics field of
logic
represents logical statements with a false base
condition
is the reason why allMatch and their
counterparts will return true for empty
sequences
can sometimes occur in real life

48. Thank you!
Bartek Żyliński
@PaskSoftware
softwaremill.com