This document provides an overview of algebra and mathematical logic. It discusses: 1) The history of algebra, from its origins in Arabic mathematics to its modern conception as the study of algebraic structures. 2) The key concepts in elementary algebra, including solving different types of equations. 3) Important terms and concepts in mathematical logic like statements, proofs, quantifiers, and methods of proof including direct proof, proof by contradiction, and proof by induction. 4) How modern abstract algebra studies algebraic structures in a broad sense.

1. Chapter 1- Preliminaries
1.1 History of Algebra
The word “algebra”-al jebr (in Arabic)
• was first used by Mohammed Al-Khwarizmi -muslim Math.
• ninth century, when taught mathematics in Baghdad.
• means “reunion”,decribes his method for collecting the terms
of an equations in order to solve it.
• Omar Khayyam, another Mathematician, defined it as
the science of solving equations.
Elementary Algebra (Classical age of Algebra)
• its central theme is clearly identified as the solving of eqs.
- method of solving linear, qudratic, cubic, quartic equations.
- 1824, Niels Abel – there does not exits any formula for
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equations degree 5 or greater.

2. Modern Age
• new varieties of algebra arose
connection with the application in math to practical problems.
- Matrix Algebra
- Bolean Algebra
- Algebra of vectors and tensors
- ~200 different kinds of algebra.
• the awareness grew
- algebra can no longer be conceived merely as the
science of solving equations.
- It had to be viewed as much more broadly as a
branch of mathematics.
revealing general principles which apply equally to
all known and all possible algebras:
* What is it that all algebras have in common?
* What trait do they share which lets us refer to all of
them as algebras? Algebraic Structure
• Abstract Algebra (Modern Algebra) -more adv. course
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The study algebraic structures.

3. 1.2 Logic and Proof
Undefined Terms
- Understand these terms and feel comfortable using them
to define new terms.
Important Terms
• Statement or Proposition
- Declarative sentence that is either true or false, but not both.
• Postulates
- Statements that are assumed to be true.
• Definition
- A precise meaning to a mathematical term.
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4. • Theorem
- A major landmark in the mathematical theory.
- Postulates and definitions are used to prove theorems.
- Once a theorem is proved to be true, it can be used.
• Lemma
- A result that is needed to prove a theorem.
• Corollary
- A result that follows immediately from a theorem.
• Example
- Is not a general result but is a particular case.
• Proof
- Mathematical argument intended to convince us that
a result is correct.
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5. Conjunction, Disjunction and Negation
Definition:
Let P and Q be statements.
i) The statement P AND Q, P ∧ Q,
is called the conjunction of P and Q.
ii) The statement P OR Q, P ∨ Q,
is called the disjunction of P and Q.
iii) The negation of P is denoted by NOT P or ~ P
Conditional and Biconditional Statement
Conditional statement: “If P then Q”, P ⇒ Q.
Biconditional statement: “P if and only if Q”, P ⇔ Q.
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6. Quantifiers
Consider a statement P(x) :
x >5
- Statement P(x) is depending on the variable x.
- Adding quantifiers can convert statement P(x) into a
statement that is either true or false.
• Universal quantifier
(∀)
P(x) is true for all values of x, denoted by
∀x, P ( x)
or
For all x, P(x).
For every x, P(x).
For each x, P(x).
P(x), for all x.
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7. • Existential quantifier (∃)
There exist an x for which P(x) is true
or
For some x, P(x).
P(x), for some x.
: ∃x, P ( x)
Example
1.
∀x ∈ R, x −1 = ( x −1)( x + x + 1)
3
2
- True or false statement? Why?
2.
∀x ∈ R, x + x − 6 = 0
2
- True or False statement? Why?
3.
∃x ∈ R, x + x − 6 = 0
2
- True or False statement? Why?
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8. Proofs
- Many mathematical theorems can be expressed
symbolically in the form of
P ⇒Q
Assumption
Or
hypothesis
Conclusion
may consists of one or more
statements.
- The theorem says that if the assumption is true than the
conclusion is true.
- How do you go about thinking up ways to prove a
theorem?
• Understand the definitions
• Try examples
• Try standard proof methods
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9. Methods of Proof ( P ⇒ Q)
1. Direct Method
• find a series of statements P1,P2,…,Pn
• verify that each of the implications below is true
P →P , P →P2 , P2 →P3 .....Pn −1 →Pn and Pn → Q
1
1
Example
An integer n is defined to be even if n = 2m for some integer m.
Show that the sum of two even integers is even.
Proof
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10. 2. Contrapositive Method
• may prove ¬Q → ¬P
Example
Proposition:
If x is a real number such that
x + 7 x < 9, then x < 1.1
3
2
Proof
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11. 3. Proof by Contradiction
• assume that P is true and not Q is true (Q is false)
• will end up with a false statement S
• Conclude that not Q must be false, i.e., Q is true
Example
Proposition:
If x is an integer and x2 is even then x is an even integer.
Proof
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12. 4. Proof by Induction
• assume that for each positive integer n,
a statement P(n) is given. If
1. P(1) is a true statement; and
2. Whenever P(k) is a true statement, then P(k+1) is also true,
• then P(n) is a true statement for every n in positive integer.
Example
Prove:
1
1
1
1
n
+
+
+ ... +
=
1• 3 3 • 5 5 • 7
(2n − 1)(2n + 1) 2n + 1
Proof
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13. 5. Proof by Counterexamples
• Sometimes a conjectured result in mathematics is not true.
• Would not be able to prove it.
• Could try to disprove it.
• The conjecture in the form of ∀x, P ( x )
• Take the negation: NOT (∀x, P ( x ))
Equivalent to:
∃x, NOT P ( x)
• Hence to disprove the statement
∀x, P ( x)
need only to find one value, say c, such that P(c) is false.
• The value c is called a counterexample to the conjecture.
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14. Example
Let x be a real number. Disprove the statement
If x2 >9 then x >3.
Solution
Remark
• To disprove the conjecture in the form of
∃x, P ( x)
cannot use counter example!!!
Its negation is equivalently in the form of ∀x, NOT
Need to show that P(x) is false for all values of x.
• To prove
P ( x)
P ⇔ Q : Prove P ⇒Q and Q ⇒P
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