An introduction to Abstract algebra in Arabic مدخل للجبر المجرد من خلال مجموعة من التعاريف We introduce binary operations and homomorphisms law. More courses will be available

1. Abstract Algebra : AN INTRODUCTON
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effective learning : See our courses on udemy.com Abstract Algebra : AN INTRODUCTON

2. Abstract Algebra
Binary Relation
E = ∅
f : E × E → E
E
b a f (a, b) E b a
a⊥b a ∧ b aTb a ∗ b
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3. Abstract Algebra
∗ = + E = R
f : R × R → R
(a, b) → a + b
N, Z, C, Q
∗ = ∪ E P(E) E = ∅
∪
∗ = ∩ E P(E) E = ∅
∩
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4. Abstract Algebra
∗
∀(a, b, c) ∈ E3
: (a ∗ b) ∗ c = a ∗ (b ∗ c)
N, Z, C, Q
∗ = ∪ E P(E) E = ∅
∪
∗ = ∩ E P(E) E = ∅
∩
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5. Abstract Algebra
∗
∀(a, b) ∈ E2
: a ∗ b = b ∗ a
N, Z, C, Q
∗ = ∪ E P(E) E = ∅
∪
∗ = ∩ E P(E) E = ∅
∩
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6. Abstract Algebra
A(R, R)
f ∈ A(R, R) ⇐⇒ f = ax + b a, b ∈ R
◦
∀f , g ∈ A(R, R) : (f ◦ g)(x) = f (g(x))
(A(R, R), ◦) ∗ = ◦
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7. Abstract Algebra
(E, ∗) E ∗ E = ∅
∃e ∈ E, ∀x ∈ E : x ∗ e = e ∗ x = x
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8. Abstract Algebra
(N, +), (Z, +), (R, +), (C, +)
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9. Abstract Algebra
(N, .), (Z, .), (R, .), (C, .)
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10. Abstract Algebra
F(X, R) = {f : X → R}
F(X, R)
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11. Abstract Algebra
E = ∅
P(E, ∪)
P(E, ∪)
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12. Abstract Algebra
E = ∅
P(E, ∩)
P(E, ∩)
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13. Abstract Algebra
(E, ∗)
R
∀(x, y) ∈ R2
: x ∗ y = xy − 3x − 3y + 4
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14. Abstract Algebra
e (E, ∗) ∗
E x
∃e ∈ E/x ∗ x = e = x ∗ x = e
E x x
E x x
∗
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15. Abstract Algebra
N, R, C, Z
∗ = +
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16. Abstract Algebra
A = (T , ◦)
∗ = ◦
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17. Abstract Algebra
E = ∅
E A = (P(E), ∪)
∗ = ∪
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18. Abstract Algebra
E = ∅
E A = (P(E), ∩)
∗ = ∩
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19. Abstract Algebra
e (E, ∗) ∗
E x E x
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20. Abstract Algebra
E S (E, ∗)
E S
∀(x, y) ∈ S2
: x ∗ y ∈ S
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21. Abstract Algebra
E E = {2n : n ∈ Z}
(Z, +), (Z, ×)
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22. Abstract Algebra
E E = {z ∈ C : |z| = 1}
(C, ×)
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23. Abstract Algebra
F(R, R) = {f : R → R}
A(R, R) = {f : R → R, f (x) = ax + b, a, b ∈ R}
(F(R, R), ◦)
F(R, R) A(R, R)
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24. Abstract Algebra
2N
f : N → 2N
a → 2a
g : N → 2N
a → 2a
f (a) + f (b) f (a + b)
g(a) + g(b) g(a + b)
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25. Abstract Algebra
⊥ ∗ (F, ⊥) (E, ∗)
F E f
∀(a, b) ∈ E2
: f (a ∗ b) = f (a)⊥f (b)
f : R∗
+ → R
x → ln(x)
f (F, ⊥) (E, ∗)
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26. Abstract Algebra
⊥ ∗ (F, ⊥) (E, ∗)
f (E) f E f (E) F E f
(F, ⊥)
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27. Abstract Algebra
⊥ ∗ (F, ⊥) (E, ∗)
f (E) f E f (E) F E f
(F, ⊥)
∀(a, b) ∈ E2
: f (a ∗ b) = f (a)⊥f (b)
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28. Abstract Algebra
⊥ ∗ (F, ⊥) (E, ∗)
(E, ∗) ∗ F E f
(F(E), ⊥). ⊥
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29. Abstract Algebra
⊥ ∗ (F, ⊥) (E, ∗)
(E, ∗) ∗ F E f
(f (E), ⊥) ⊥
∀(a, b) ∈ E2
: f (a ∗ b) = f (a)⊥f (b)
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30. Abstract Algebra
⊥ ∗ (F, ⊥) (E, ∗)
e (E, ∗) F E f
(f (E), ⊥) e = f (e)
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31. Abstract Algebra
⊥ ∗ (F, ⊥) (E, ∗)
e ∗ F E f
y = f (x) (E, ∗) x x
y = f (x ) (f (E), ⊥)
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