WEBVTT

1
00:00:00.000 --> 00:00:06.000
This overview explains how four Boolean cases
form an operator correspondence matrix,

2
00:00:06.000 --> 00:00:10.033
and why sixteen such matrices are possible.

3
00:00:10.033 --> 00:00:18.200
A declared axis order makes every cell interpretable
and prevents confusing one case with a whole function.

4
00:00:18.500 --> 00:00:23.113
For these operator matrices, use true-first axes.

5
00:00:23.113 --> 00:00:27.946
The upper row means A one and the lower row means A zero.

6
00:00:27.946 --> 00:00:32.826
The left column means B one and the right column means B zero.

7
00:00:32.826 --> 00:00:37.872
And has one only at the upper-left cell, where both inputs are one.

8
00:00:38.172 --> 00:00:41.160
A transpose changes a row into a column.

9
00:00:41.160 --> 00:00:44.426
Take column one zero and row zero one.

10
00:00:44.426 --> 00:00:55.296
Multiply each row entry by each column entry using and: one and zero is zero, one and one is one, zero and zero is zero, and zero and one is zero.

11
00:00:55.296 --> 00:00:59.432
The resulting array is zero one, zero zero.

12
00:00:59.432 --> 00:01:08.361
This column-times-row construction is called an outer product; a single-cell building piece is also called a basis dyad.

13
00:01:09.111 --> 00:01:13.294
Each of the four cells can independently be zero or one.

14
00:01:13.294 --> 00:01:17.767
Four binary choices give sixteen possible complete arrays.

15
00:01:17.767 --> 00:01:21.044
This counts Boolean functions of two inputs.

16
00:01:21.044 --> 00:01:27.313
The four input assignments and the sixteen possible functions are different counts.

17
00:01:27.313 --> 00:01:27.316
The four input assignments and the sixteen possible functions are different counts.

18
00:01:27.316 --> 00:01:34.116
X labels the rows and Y labels the columns,
with true before false on each axis.

19
00:01:34.116 --> 00:01:40.550
The four cases are 11, 10, 01, 00.

20
00:01:40.550 --> 00:01:43.350
One cell records one case.

21
00:01:43.350 --> 00:01:48.250
All four output bits form one operator correspondence matrix.

22
00:01:48.250 --> 00:01:52.350
Bracket Theta names the chosen operator matrix.

23
00:01:52.350 --> 00:01:59.450
AND returns one only when both inputs are true, giving 1000 in that order.

24
00:01:59.450 --> 00:02:05.016
True is represented by the column one, zero; false by zero, one.

25
00:02:05.016 --> 00:02:08.883
Transposing a column ket makes a row bra.

26
00:02:08.883 --> 00:02:14.983
The outer product of ket one and bra
zero places a single one at the top right.

27
00:02:14.983 --> 00:02:19.416
The four basis dyads select the four positions.

28
00:02:19.416 --> 00:02:23.516
Each of four output positions has two choices.

29
00:02:23.516 --> 00:02:28.883
2⁴ gives sixteen possible two-input functions.

30
00:02:28.883 --> 00:02:35.450
Every small grid in this table is a complete
function under the same true-first axes.

31
00:02:35.450 --> 00:02:39.450
Combining the required basis dyads with exclusive-or builds a matrix:

32
00:02:39.450 --> 00:02:44.750
AND uses one, OR three, and XOR the two off-diagonal dyads.

33
00:02:44.750 --> 00:02:48.516
Equivalence has ones on the agreement diagonal.

34
00:02:48.516 --> 00:02:55.250
For X implies Y, only the case X true and Y false fails.

35
00:02:55.250 --> 00:03:00.550
Reversing the operands moves the failing case to the other off-diagonal cell.

36
00:03:00.550 --> 00:03:04.600
The axis labels keep that distinction explicit.

37
00:03:04.600 --> 00:03:04.616
The axis labels keep that distinction explicit.

38
00:03:04.916 --> 00:03:09.150
Work a second complete case with A 0, B 1.

39
00:03:09.150 --> 00:03:12.650
Keep this assignment fixed throughout the calculation.

40
00:03:12.650 --> 00:03:16.916
Each symbol retains the meaning already introduced; a new value does not

41
00:03:16.916 --> 00:03:20.150
create a new logical rule.

42
00:03:20.150 --> 00:03:25.316
A zero selects the lower row, while B one selects the left column.

43
00:03:25.316 --> 00:03:29.483
The implication matrix contains one at that intersection.

44
00:03:29.483 --> 00:03:34.150
Reverse the operand roles: one implies zero is false.

45
00:03:34.150 --> 00:03:38.350
Tracking this asymmetric pair exposes a transposition error that a

46
00:03:38.350 --> 00:03:42.216
symmetric and matrix could conceal.

47
00:03:42.216 --> 00:03:45.950
This case establishes a specific contrast.

48
00:03:45.950 --> 00:03:49.116
Operand order changes implication.

49
00:03:49.116 --> 00:03:51.750
Axis labels are required.

50
00:03:51.750 --> 00:03:56.416
Keep the input, intermediate values, and conclusion together when comparing

51
00:03:56.416 --> 00:03:58.616
it with the previous case.

52
00:03:58.616 --> 00:04:05.250
The calculated answer belongs to this complete assignment under the stated rule.

53
00:04:05.583 --> 00:04:09.583
Or accepts the true-true cell and both mixed cells.

54
00:04:09.583 --> 00:04:13.350
Exclusive-or accepts only the two mixed cells.

55
00:04:13.350 --> 00:04:17.916
Compare their upper-left entries: or returns one when both inputs are true, while

56
00:04:17.916 --> 00:04:20.850
exclusive-or returns zero.

57
00:04:20.850 --> 00:04:27.083
That single cell captures the difference between inclusive and exclusive alternatives.

58
00:04:27.083 --> 00:04:32.450
Equivalence accepts both equal-input cells, true true and false false.

59
00:04:32.450 --> 00:04:36.650
Its matrix has ones on the diagonal in this declared order.

60
00:04:36.650 --> 00:04:41.616
The name describes agreement of the two input values; it does not mean every expression

61
00:04:41.616 --> 00:04:46.250
containing this operator is equivalent to another expression.

62
00:04:46.250 --> 00:04:50.583
Implication fails when A is true and B is false.

63
00:04:50.583 --> 00:04:53.083
The reversed input gives one.

64
00:04:53.083 --> 00:04:56.983
If we exchange operand roles, we must also exchange row and column

65
00:04:56.983 --> 00:04:59.783
roles by transposing the matrix.

66
00:04:59.783 --> 00:05:03.850
Symmetric operators can hide an ordering mistake, so implication is a

67
00:05:03.850 --> 00:05:07.450
useful check of orientation.

68
00:05:07.450 --> 00:05:07.461
Four cases describe one two-input function.

69
00:05:07.461 --> 00:05:11.883
Four cases describe one two-input function.

70
00:05:11.883 --> 00:05:17.467
Choosing their four outputs gives sixteen possible functions.