This presentation gives a very simple introduction to deep learning in addition to a step-by-step example showing how to solve the XOR non-linear problem using multi-layer artificial neural networks that has both input, hidden, and output layers. Deep learning is based on artificial neural networks and it aims to analyze large amounts of data that are not easily analyzed using conventional models. It creates a large neural network with several hidden layers and several neurons within each layer and usually may take days for its learning. Many beginners in artificial neural networks have a problem in understanding how hidden layers are useful and what is the best number of hidden layers and best number of neurons or nodes within each layer. أحمد فوزي جاد Ahmed Fawzy Gad قسم تكنولوجيا المعلومات Information Technology (IT) Department كلية الحاسبات والمعلومات Faculty of Computers and Information (FCI) جامعة المنوفية, مصر Menoufia University, Egypt Teaching Assistant/Demonstrator ahmed.fawzy@ci.menofia.edu.eg : AFCIT http://www.afcit.xyz YouTube https://www.youtube.com/channel/UCuewOYbBXH5gwhfOrQOZOdw Google Plus https://plus.google.com/u/0/+AhmedGadIT SlideShare https://www.slideshare.net/AhmedGadFCIT LinkedIn https://www.linkedin.com/in/ahmedfgad/ ResearchGate https://www.researchgate.net/profile/Ahmed_Gad13 Academia https://menofia.academia.edu/Gad Google Scholar https://scholar.google.com.eg/citations?user=r07tjocAAAAJ&hl=en Mendelay https://www.mendeley.com/profiles/ahmed-gad12/ ORCID https://orcid.org/0000-0003-1978-8574 StackOverFlow http://stackoverflow.com/users/5426539/ahmed-gad Twitter https://twitter.com/ahmedfgad Facebook https://www.facebook.com/ahmed.f.gadd Pinterest https://www.pinterest.com/ahmedfgad/

1. Brief Introduction to Deep Learning +
Solving XOR using ANN
MENOUFIA UNIVERSITY
FACULTY OF COMPUTERS AND INFORMATION
‫المنوفية‬ ‫جامعة‬
‫الحاسبات‬ ‫كلية‬‫والمعلومات‬
‫المنوفية‬ ‫جامعة‬
Ahmed Fawzy Gad
ahmed.fawzy@ci.menofia.edu.eg

2. Classification Example
BA
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3. Neural Networks
Input Hidden Output
BA
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4. Neural Networks
BA
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9. 0
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12. =
+0
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Network Architecture!!

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Input OutputHidden

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Input OutputHidden

19. 0
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Input OutputHidden

20. 0
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BA
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Input Output

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BA
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Input Output

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Input Output

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Input Output
A
B

24. 0
0.2
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1.2
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BA
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Input Output
A
B

25. 0
0.2
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1.2
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Input Output
A
B

26. 0
0.2
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BA
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Input Output
A
B

27. 0
0.2
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BA
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Input Output
A
B
1/0

28. 0
0.2
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Input Output
A
B
1/0
𝑾 𝟏
𝑾 𝟐

29. 0
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+ 0
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0
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32. 0
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Input Output
A
B
1/0
𝑾 𝟑
𝑾 𝟒

33. 0
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A
B
1/0
𝑾 𝟑
𝑾 𝟒
A
B
1/0
𝑾 𝟏
𝑾 𝟐
0
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34. 0
0.2
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𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
A
B
𝑾 𝟑
𝑾 𝟒

35. A
B
𝑾 𝟏
𝑾 𝟐
A
B
𝑾 𝟑
𝑾 𝟒
𝑾 𝟓
𝑾 𝟔

36. 𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
A
B
𝑾 𝟑
𝑾 𝟒
𝑾 𝟓
𝑾 𝟔
A 𝑾 𝟏
𝑾 𝟐
B
𝑾 𝟑
𝑾 𝟒

37. 𝑾 𝟓
𝑾 𝟔
A 𝑾 𝟏
𝑾 𝟐
B
𝑾 𝟑
𝑾 𝟒

39. .
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40. .
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42. .
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43. .
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44. .
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45. .
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46. .
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52. Input Output
BA
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0
0.2
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0.8
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1.2
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Hidden
Weights=𝑾𝒊
𝑾 𝟓
𝑾 𝟔
1/0
𝒀𝒋
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

53. Activation Function
Output
𝒀𝒋
BA
01
1
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Input Hidden
1/0
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

54. Activation Function
Output
𝒀𝒋
BA
01
1
10
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0
11
Input Hidden
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

55. Activation Function
Output
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

56. Activation Function
Components
Output
𝒀𝒋
BA
01
1
10
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0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

57. Activation Function
Inputs
Output
s
𝒀𝒋
BA
01
1
10
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0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

58. Activation Function
Inputs
Output
s
𝒀𝒋
BA
01
1
10
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0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
s=SOP(𝑿𝒊, 𝑾𝒊)

59. Activation Function
Inputs
Output
s
𝑿𝒊=Inputs 𝑾𝒊=Weights
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
s=SOP(𝑿𝒊, 𝑾𝒊)

60. Activation Function
Inputs
Output
s
𝑿𝒊=Inputs 𝑾𝒊=Weights
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
s=SOP(𝑿𝒊, 𝑾𝒊)

61. 𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
Activation Function
Inputs
Output
s
𝑿𝒊=Inputs 𝑾𝒊=Weights
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
S= 𝟏
𝒎
𝑿𝒊 𝑾𝒊
s=SOP(𝑿𝒊, 𝑾𝒊)
1/0

62. Activation Function
Inputs
Output
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
S= 𝟏
𝒎
𝑿𝒊 𝑾𝒊
1/0
Each Hidden/Output Layer
Neuron has its SOP.

63. Activation Function
Inputs
Output
s
𝑿 𝟏
𝑿 𝟐
𝑺 𝟏=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑)
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
S= 𝟏
𝒎
𝑿𝒊 𝑾𝒊
1/0

64. Activation Function
Inputs
Output
s
𝑿 𝟏
𝑿 𝟐
𝑺 𝟐=(𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒)
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
S= 𝟏
𝒎
𝑿𝒊 𝑾𝒊
1/0

65. Activation Function
Inputs
Output
s
𝑿 𝟏
𝑿 𝟐
𝑺 𝟑=(𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔)
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
S= 𝟏
𝒎
𝑿𝒊 𝑾𝒊
1/0

66. Activation Function
Outputs
Output
F(s)s
𝑿 𝟏
𝑿 𝟐
Class Label
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

67. Activation Function
Outputs
Output
F(s)s
𝑿 𝟏
𝑿 𝟐
Class Label
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Input Hidden
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

68. Activation Functions
Piecewise
Linear Sigmoid Binary

69. Activation Functions
Which activation function to use?
Outputs
Class
Labels
Activation
Function
TWO Class
Labels
TWO
Outputs
One that gives two outputs.
Which activation function to use?
𝑪𝒋𝒀𝒋
BA
01
1
10
00
0
11
BA
01
1 10
00
0 11

70. Activation Functions
Piecewise
Linear Sigmoid BinaryBinary

71. Activation Function
Output
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
BA
01
1
10
00
0
11
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
1/0
Input Hidden
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

72. Bias
Input Output
BA
01
1
10
00
0
11
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

73. Bias
Hidden Layer Neurons
Input Output
BA
01
1
10
00
0
11
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
+1
𝒃 𝟏
+1
𝒃 𝟐
1/0

74. Bias
Output Layer Neurons
Input Output
BA
01
1
10
00
0
11
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
+1
𝒃 𝟑
1/0
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

75. All Bias Values
Input Output
BA
01
1
10
00
0
11
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
+1
𝒃 𝟏
+1
𝒃 𝟐
1/0
+1
𝒃 𝟑
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

76. Bias
Add Bias to SOP
Input Output
BA
01
1
10
00
0
11
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
+1
𝒃 𝟏
+1
𝒃 𝟐
𝑺 𝟏=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑)
𝑺 𝟐=(𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒)
𝑺 𝟑=(𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔)
1/0
+1
𝒃 𝟑
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

77. Bias
Add Bias to SOP
Input Output
BA
01
1
10
00
0
11
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
+1
𝒃 𝟏
+1
𝒃 𝟐
𝑺 𝟏=(𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑)
1/0
+1
𝒃 𝟑
𝑺 𝟏=(+𝟏𝒃 𝟏+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑)
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

78. Bias
Add Bias to SOP
Input Output
BA
01
1
10
00
0
11
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
+1
𝒃 𝟏
+1
𝒃 𝟐
𝑺 𝟐=(𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒)
1/0
+1
𝒃 𝟑
𝑺 𝟐=(+𝟏𝒃 𝟐+𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒)
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

79. Bias
Add Bias to SOP
Input Output
BA
01
1
10
00
0
11
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
+1
𝒃 𝟏
+1
𝒃 𝟐
𝑺 𝟑=(𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔)
1/0
+1
𝒃 𝟑
𝑺 𝟑=(+𝟏𝒃 𝟑+𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔)
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

80. Learning Rate
𝟎 ≤ η ≤ 𝟏
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
+1
𝒃 𝟏
+1
𝒃 𝟐
1/0
+1
𝒃 𝟑
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒

81. Other Parameters
Step n
𝒏 = 𝟎, 𝟏, 𝟐, …
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
+1
𝒃 𝟏
+1
𝒃 𝟐
1/0
+1
𝒃 𝟑
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+…)
𝟎 ≤ η ≤ 𝟏
𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, …)
W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, …)

82. Other Parameters
Desired Output 𝒅𝒋
𝒏 = 𝟎, 𝟏, 𝟐, …
𝒅 𝒏 =
𝟏, 𝒙 𝒏 𝒃𝒆𝒍𝒐𝒏𝒈𝒔 𝒕𝒐 𝑪𝟏 (𝟏)
𝟎, 𝒙 𝒏 𝒃𝒆𝒍𝒐𝒏𝒈𝒔 𝒕𝒐 𝑪𝟐 (𝟎)
BA
01
1
10
00
0
11
F(s)s
𝑿 𝟏
𝑿 𝟐
bin
𝒀𝒋
+1
𝒃 𝟏
+1
𝒃 𝟐
1/0
+1
𝒃 𝟑
𝑾 𝟓
𝑾 𝟔
A
B
𝑾 𝟏
𝑾 𝟐
𝑾 𝟑
𝑾 𝟒
s=(𝑿 𝟎 𝑾 𝟎+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟐+…)
𝟎 ≤ η ≤ 𝟏
𝑿(𝒏)=(𝑿 𝟎, 𝑿 𝟏,𝑿 𝟐, …)
W(𝒏)=(𝑾 𝟎, 𝑾 𝟏,𝑾 𝟐, …)

83. Neural Networks Training Steps
Weights Initialization
Inputs Application
Sum of Inputs-Weights Products
Activation Function Response Calculation
Weights Adaptation
Back to Step 2
1
2
3
4
5
6

84. Regarding 5th Step: Weights Adaptation
• If the predicted output Y is not the same as the desired output d,
then weights are to be adapted according to the following equation:
𝑾 𝒏 + 𝟏 = 𝑾 𝒏 + η 𝒅 𝒏 − 𝒀 𝒏 𝑿(𝒏)
Where
𝑾 𝒏 = [𝒃 𝒏 , 𝑾 𝟏(𝒏), 𝑾 𝟐(𝒏), 𝑾 𝟑(𝒏), … , 𝑾 𝒎(𝒏)]

85. Neural Networks
Training Example
Step n=0
• In each step in the solution, the parameters of the neural network
must be known.
• Parameters of step n=0:
η = .001
𝑋 𝑛 = 𝑋 0 = +1, +1, +1,1, 0
𝑊 𝑛 = 𝑊 0 = 𝑏1, 𝑏2, 𝑏3, 𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6
= −1.5, −.5, −.5, 1, 1, 1, 1, −2, 1
𝑑 𝑛 = 𝑑 0 = 1
BA
01
1 => 1
10
00
0 => 0
11

86. Neural Networks
Training Example
Step n=0
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin
BA
01
1 => 1
10
00
0 => 0
11

87. Neural Networks
Training Example
Step n=0 – SOP – 𝑺 𝟏
𝑺 𝟏=(+𝟏𝒃 𝟏+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑)
=+1*-1.5+1*1+0*1
=-.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

88. Neural Networks
Training Example
Step n=0 – Output – 𝑺 𝟏
𝒀 𝑺 𝟏 =
= 𝑩𝑰𝑵 𝑺 𝟏
= 𝑩𝑰𝑵 −. 𝟓
= 𝟎
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
𝟎, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

89. Neural Networks
Training Example
Step n=0 – SOP – 𝑺 𝟐
𝑺 𝟐=(+𝟏𝒃 𝟐+𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒)
=+1*-.5+1*1+0*1
=.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

90. Neural Networks
Training Example
Step n=0 – Output – 𝑺 𝟐
𝒀 𝑺2 =
= 𝑩𝑰𝑵 𝑺2
= 𝑩𝑰𝑵 . 𝟓
= 1
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
−𝟏, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

91. Neural Networks
Training Example
Step n=0 – SOP – 𝑺 𝟑
𝑺 𝟑=(+𝟏𝒃 𝟑+𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔)
=+1*-.5+0*-2+1*1
=.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

92. Neural Networks
Training Example
Step n=0 – Output – 𝑺 𝟑
𝒀 𝑺3 =
= 𝑩𝑰𝑵 𝑺3
= 𝑩𝑰𝑵 . 𝟓
= 1
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
𝟎, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

93. Neural Networks
Training Example
Step n=0 - Output
𝒀 𝒏 = 𝒀 𝟎 = 𝒀 𝑺3
= 1
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

94. Neural Networks
Training Example
Step n=0
Predicted Vs. Desired
𝒀 𝒏 = 𝒀 𝟎 = 1
𝐝 𝒏 = 𝒅 𝟎 = 1
∵ 𝒀 𝒏 = 𝒅 𝒏
∴ Weights are Correct.
No Adaptation
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

95. Neural Networks
Training Example
Step n=1
• In each step in the solution, the parameters of the neural network
must be known.
• Parameters of step n=1:
η = .001
𝑋 𝑛 = 𝑋 1 = +1, +1, +1,0, 1
𝑊 𝑛 = 𝑊 1 = 𝑊 0 = 𝑏1, 𝑏2, 𝑏3, 𝑤1, 𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6
= −1.5, −.5, −.5, 1, 1, 1, 1, −2, 1
𝑑 𝑛 = 𝑑 1 = +1
BA
01
1 => 1
10
00
0 => 0
11

96. Neural Networks
Training Example
Step n=1
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin
BA
01
1 => 1
10
00
0 => 0
11

97. Neural Networks
Training Example
Step n=1 – SOP – 𝑺 𝟏
𝑺 𝟏=(+𝟏𝒃 𝟏+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑)
=+1*-1.5+0*1+1*1
=-.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

98. Neural Networks
Training Example
Step n=1 – Output – 𝑺 𝟏
𝒀 𝑺 𝟏 =
= 𝑩𝑰𝑵 𝑺 𝟏
= 𝑩𝑰𝑵 −. 𝟓
= 𝟎
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
𝟎, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

99. Neural Networks
Training Example
Step n=1 – SOP – 𝑺 𝟐
𝑺 𝟐=(+𝟏𝒃 𝟐+𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒)
=+1*-.5+0*1+1*1
=.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

100. Neural Networks
Training Example
Step n=1 – Output – 𝑺 𝟐
𝒀 𝑺2 =
= 𝑩𝑰𝑵 𝑺2
= 𝑩𝑰𝑵 . 𝟓
= 1
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
−𝟏, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

101. Neural Networks
Training Example
Step n=1 – SOP – 𝑺 𝟑
𝑺 𝟑=(+𝟏𝒃 𝟑+𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔)
=+1*-.5+0*-2+1*1
=.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

102. Neural Networks
Training Example
Step n=1 – Output – 𝑺 𝟑
𝒀 𝑺3 =
= 𝑩𝑰𝑵 𝑺3
= 𝑩𝑰𝑵 . 𝟓
= 1
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
𝟎, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

103. Neural Networks
Training Example
Step n=1 - Output
𝒀 𝒏 = 𝒀 𝟏 = 𝒀 𝑺3
= 1
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

104. Neural Networks
Training Example
Step n=1
Predicted Vs. Desired
𝒀 𝒏 = 𝒀 𝟏 = 1
𝐝 𝒏 = 𝒅 𝟏 = 1
∵ 𝒀 𝒏 = 𝒅 𝒏
∴ Weights are Correct.
No Adaptation
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−2
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

105. Neural Networks
Training Example
Step n=2
• In each step in the solution, the parameters of the neural network
must be known.
• Parameters of step n=2:
η = .001
𝑋 𝑛 = 𝑋 2 = +1, +1, +1,0, 0
𝑊 𝑛 = 𝑊 2 = 𝑊 1 = 𝑏1, 𝑏2, 𝑏3, 𝑤1, 𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6
= −1.5, −.5, −.5, 1, 1, 1, 1, −2, 1
𝑑 𝑛 = 𝑑 2 = 0
BA
01
1 => 1
10
00
0 => 0
11

106. Neural Networks
Training Example
Step n=2
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin
BA
01
1 => 1
10
00
0 => 0
11

107. Neural Networks
Training Example
Step n=2 – SOP – 𝑺 𝟏
𝑺 𝟏=(+𝟏𝒃 𝟏+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑)
=+1*-1.5+0*1+0*1
=-1.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

108. Neural Networks
Training Example
Step n=2 – Output – 𝑺 𝟏
𝒀 𝑺 𝟏 =
= 𝑩𝑰𝑵 𝑺 𝟏
= 𝑩𝑰𝑵 −𝟏. 𝟓
= 𝟎
𝒃𝒊n 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
𝟎, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

109. Neural Networks
Training Example
Step n=2 – SOP – 𝑺 𝟐
𝑺 𝟐=(+𝟏𝒃 𝟐+𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒)
=+1*-.5+0*1+0*1
=-.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

110. Neural Networks
Training Example
Step n=2 – Output – 𝑺 𝟐
𝒀 𝑺2 =
= 𝑺𝑮𝑵 𝑺2
= 𝑺𝑮𝑵 −. 𝟓
=0
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
−𝟏, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

111. Neural Networks
Training Example
Step n=2 – SOP – 𝑺 𝟑
𝑺 𝟑=(+𝟏𝒃 𝟑+𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔)
=+1*-.5+0*-2+0*1
=-.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

112. Neural Networks
Training Example
Step n=2 – Output – 𝑺 𝟑
𝒀 𝑺3 =
= 𝑩𝑰𝑵 𝑺3
= 𝑩𝑰𝑵 −. 𝟓
= 𝟎
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
𝟎, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

113. Neural Networks
Training Example
Step n=2 - Output
𝒀 𝒏 = 𝒀 𝟐 = 𝒀 𝑺3
= 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

114. Neural Networks
Training Example
Step n=2
Predicted Vs. Desired
𝒀 𝒏 = 𝒀 𝟐 = 𝟎
𝐝 𝒏 = 𝒅 𝟐 = 𝟎
∵ 𝒀 𝒏 = 𝒅 𝒏
∴ Weights are Correct.
No Adaptation
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

115. Neural Networks
Training Example
Step n=3
• In each step in the solution, the parameters of the neural network
must be known.
• Parameters of step n=3:
η = .001
𝑋 𝑛 = 𝑋 3 = +1, +1, +1,1, 1
𝑊 𝑛 = 𝑊 3 = 𝑊 2 = 𝑏1, 𝑏2, 𝑏3, 𝑤1, 𝑤1, 𝑤2, 𝑤3, 𝑤4, 𝑤5, 𝑤6
= −1.5, −.5, −.5, 1, 1, 1, 1, −2, 1
𝑑 𝑛 = 𝑑 3 = 0
BA
01
1 => 1
10
00
0 => 0
11

116. Neural Networks
Training Example
Step n=3
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin
BA
01
1 => 1
10
00
0 => 0
11

117. Neural Networks
Training Example
Step n=3 – SOP – 𝑺 𝟏
𝑺 𝟏=(+𝟏𝒃 𝟏+𝑿 𝟏 𝑾 𝟏+𝑿 𝟐 𝑾 𝟑)
=+1*-1.5+1*1+1*1
=.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

118. Neural Networks
Training Example
Step n=3 – Output – 𝑺 𝟏
𝒀 𝑺 𝟏 =
= 𝑩𝑰𝑵 𝑺 𝟏
= 𝑩𝑰𝑵 . 𝟓
= 1
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
𝟎, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

119. Neural Networks
Training Example
Step n=3 – SOP – 𝑺 𝟐
𝑺 𝟐=(+𝟏𝒃 𝟐+𝑿 𝟏 𝑾 𝟐+𝑿 𝟐 𝑾 𝟒)
=+1*-.5+1*1+1*1
=1.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

120. Neural Networks
Training Example
Step n=3 – Output – 𝑺 𝟐
𝒀 𝑺2 =
= 𝑩𝑰𝑵 𝑺2
= 𝑩𝑰𝑵 𝟏. 𝟓
= 1
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
−𝟏, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

121. Neural Networks
Training Example
Step n=3 – SOP – 𝑺 𝟑
𝑺 𝟑=(+𝟏𝒃 𝟑+𝑺 𝟏 𝑾 𝟓+𝑺 𝟐 𝑾 𝟔)
=+1*-.5+1*-2+1*1
=-1.5
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

122. Neural Networks
Training Example
Step n=3 – Output – 𝑺 𝟑
𝒀 𝑺3 =
= 𝑩𝑰𝑵 𝑺3
= 𝑩𝑰𝑵 −𝟏. 𝟓
= 𝟎
𝒃𝒊𝒏 𝒔 =
+𝟏, 𝒔 ≥ 𝟎
𝟎, 𝒔 < 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

123. Neural Networks
Training Example
Step n=3 - Output
𝒀 𝒏 = 𝒀 𝟑 = 𝒀 𝑺3
= 𝟎
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

124. Neural Networks
Training Example
Step n=3
Predicted Vs. Desired
𝒀 𝒏 = 𝒀 𝟑 = 𝟎
𝐝 𝒏 = 𝒅 𝟑 = 𝟎
∵ 𝒀 𝒏 = 𝒅 𝒏
∴ Weights are Correct.
No Adaptation
BA
01
1 => 1
10
00
0 => 0
11
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin

125. Final Weights
s
𝑿 𝟏
𝑿 𝟐
𝒀𝒋
+1
−𝟏. 𝟓
+1
−. 𝟓
1/0
+1
−. 𝟓
−𝟐
+𝟏
A
B
+𝟏
+𝟏
+𝟏
+𝟏
bin
Current weights predicted
the desired outputs.