MA1006 ALGEBRA – List 1
1. Read the wikipedia article about mathematics, algebra and logic.
2. Read the wikipedia article about truth table. Memorise the truth table for: negation,
conjucntion, disjunction, implication and equality (aka equivalence).
3. Which of the following sentences are statements? Which of the statements are true?
a. Let x be a natural number.
b. A natural number x is divisible by three.
c. For any even natural number x its square x 2 is divisible by four.
d.
x is a natural number.
e. Every integer is positive.
f. It is not true that this sentence is true.
g. There are two people in Aberdeen having the same amount of hair on their heads.
h. The Algebra lecturer is cool.
i. There are infinitely many prime numbers.
j. If 1 = 2 then the Earth crashed with the Moon last night.
4. Read sections 1 to 7 in the address
http://people.hofstra.edu/Stefan_Waner/RealWorld/logic/logicintro.html
and solve the examples and exercises proposed there.
5. Determine if the following statements are true or false. Prove the true ones and provide a
counterexample for each of the false ones.
a. If n is a natural number then n 2 is divisible by four.
b. If a natural number n is divisible by four then it is a square of a natural number.
c. There exists infinitely many prime numbers.
d. A natural number n is odd if and only if n 2 is odd.
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e. The number 3 is not a rational number.
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f. For every natural number n we have that ni=1 i = n(n−1)
.
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g. For every natural number n we have that i =1 (2i − 1) = n 2 .
6. Write the negation of each of the following statements.
a. For every integer n there exists an integer m such that m ≥ n .
b. If x is an even natural number then x 2 is divisible by three.
c. For every ² > 0 there exists a natural number m such that |a n − a| < ² for every n > m .
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d. For every x 0 ∈ R and every ² > 0 there exists δ > 0 such that |F(x) − F(x 0 )| < ² for every x
such that |x − x 0 | < δ.
7. Prove that the following statement is true. For each natural number k ≥ 2 and all n ≥ 0, k n −1
is divisible by k − 1.
8. The Fibonacci sequence is a sequence of natural numbers (roughly speaking, a list of natural
numbers a 1 , a 2 , a 3 , . . . that continues to infinity) that appears in nature in many different
ways. You can read about it in
http://en.wikipedia.org/wiki/Fibonacci_number
The Fibonacci sequence is defined as follows. F0 = 0, F1 = 1 and, for n ≥ 2,
Fn = Fn−1 + Fn−2 .
Prove the following properties of the Fibonacci sequence.
a. For all n ≥ 3, 1 <
Fn+1
Fn
< 2.
b. For all n ≥ 1, F4n is divisible by 3.
9. Prove that if a is the side length of a cube and d is the length of a diagonal joining opposite
corners of the cube, then d 2 = 3a 2 .
Summary of Week 1:
• General introduction to the context of mathematics in science and and all human activity.
• Introducing some of the mathematical notation.
• Elements of logic: statement, proof, counterexample, quantifiers; the truth table for negation, conjunction, disjunction, implication and equivalence. Many examples.
• Methods of proofs: direct, by contradiction, induction. Many examples.
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