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Mathematics 1 Dersi 1. Ünite Sorularla Öğrenelim

Sets And Numbers

1. Soru

What is a set?

Cevap

A set is a well-defined collection of elements, that is, it must be clear whether a given element
belongs to the set or not.


2. Soru

How are sets denoted by?

Cevap

Usually, sets are denoted by the capital letters A, B, C, etc., and elements of the sets by lower case letters a, b, c etc. If A is a set and a is an element of A we use the notation a ? A, if b is not an element of A then we write b ? A.


3. Soru

What are empty sets?

Cevap

The empty set is unique.The empty set Ø is a subset of any set. Note that the empty set is a subset of any set A and A is its own subset: Ø ? A, A ? A. . If the set A is not a subset of the set D, this relation is written as A ? D..


4. Soru

How can you define the union of the sets?

Cevap

The set of all elements of the sets A and B is called the union of the sets A and B and is denoted by A ? B.Union is the act of combining two sets together into a single set.


5. Soru

If A = {3, 5, 8, 10}, B = {4, 5, 9}. Then what is A \ B?

Cevap

The set of elements A which are not in B is called the difference between A and B and is denoted by A \ B. A = {3, 5, 8, 10}, B = {4, 5, 9}. Then A \ B = {3, 8, 10}.


6. Soru

Ac = U \ A = {x| x ? U and x ? A} What can be said about this equation?

Cevap

Usually the sets that we deal with are subsets of some ambient set. Such a set is called a universal set and is denoted by U. In other words, U is the universal set if all the sets under examination are subsets of U. The difference U \ A is called the complement of A and is denoted by Ac . That is,
Ac = U \ A = {x| x ? U and x ? A}


7. Soru

If A = {1, 2, 3, 5, 8}, B = {2, 3, 10, 11}. Then A ? B =?

Cevap

The intersection of two sets A and B, written A ? B is the set consisting of the elements of both A
and B. Thus, A ? B = {x| x ? A and x ? B}

A = {1, 2, 3, 5, 8}, B = {2, 3, 10, 11}. Then A ? B = {2, 3}.


8. Soru

When the intersection of two sets is the empty , what does it called?

Cevap

Two sets A and B are called disjoint if A ? B = Ø, that is their intersection is the empty set.


9. Soru

What is the difference  between finite and infinite set?

Cevap

A set A is called finite if it consists of a finite number of elements. Otherwise it is called an
infinite set.The set A = {1 ,3, 5} is finite, whereas the set B = {1, 3, 5, 7, …} is infinite.
For a finite set A, the number of elements in this set is denoted by s(A). For A = {1, 3, 5} we have s(A) = 3. The following equality is true:
s(A ? B) = s(A) + s(B) –s(A ? B)


10. Soru

If A = {3, 5}, B = {1, 2, 3, 5}, C = {1, 2, 4, 6}. Then what is A ? B, A ? B,B ? C, A ? C=? 

Cevap

A = {3, 5}, B = {1, 2, 3, 5}, C = {1, 2, 4, 6}. Then
A ? B, A ? B = A = {3, 5},
B ? C = {1, 2, 3, 4, 5, 6},
A ? C = Ø


11. Soru

If A = {1, 2, 3} then What are the subsets of A?

Cevap

Ø, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.
The set A has 2x2x2 = 8 subsets.


12. Soru

How would you define natural numbers?

Cevap

The most familiar number set is the natural numbers, denoted by N= {1, 2, 3, 4, ...}Every natural number is an integer. Every integer is a rational number. We do not consider the zero as a natural number. The sum of two natural numbers is again a natural number, whereas the difference might not be a natural number.


13. Soru

What is a rational number?

Cevap

A rational number is defined as a quotient of two integers with nonzero denominator and is
denoted by Q. A rational number is a fraction m/n where m and n are integers and n ? 0. Every rational number has an infinite number of representations by fractions. Rational numbers with denominators 10, 100, 1000, ... have special representations, named decimal representations. Every rational number has a finite or repeated infinitive decimal representation by using decimal fractions. 


14. Soru

What is a real number which is not a rational?

Cevap

A real number which is not a rational is an irrational number.Every irrational number has rational numbers arbitrarily close to it. The union of the sets of rational and irrational numbers is called the set of real numbers and is denoted by R .


15. Soru

Assume that, a < 0, 4a = 3b, b = 2c. Write the numbers a, b and c in increasing order.

Cevap

3b = 4a, b = 4/3 · a. Since a < 0 then b < a. c = b/2 = 4/3· 2 · a = 2/3 · a.

Therefore c > a, and b < a < c.


16. Soru

What is it called when you multiply a real number by itself repeatedly?

Cevap

Powers are used when we multiply a real number by itself repeatedly. For a ? R and n ? N , we define a to the power n.For a ? 0, the zero exponent and the negative exponents are defined as follows:a0 =1 a-n = 1/an . If n is an even natural number then nth root of a, an , is defined
only for nonnegative numbers a. If n is odd then an is defined for all numbers a.


17. Soru

What are the properties of the powers?

Cevap

(a ·b)n = an ·bn

(a/b)= an/bn   (b? 0)

am · an = am+n

(am)n=am+n


18. Soru

What are Intervals?

Cevap

Intervals are important subsets of the real numbers. Given two real numbers a and b with a < b the set {x| x ? R and a ? x ? b}is called a closed interval and is denoted by [a, b]. Similarly, the half-open intervals (a, b], [a, b) are defined as (a, b] = {x| x ? R and a < x ? b}, [a, b) = {x| x ? R and a ? x < b}.The intervals, defined above are finite intervals. Using the symbols ? (plus infinity) and –? (minus infinity) unbounded intervals can be defined, namely

(a, ?) = {x| x ? R and x > a},

a, ?) = {x| x ? R and x ? a},

(–?, b) = {x| x ? R and x < b},
(–?, b] = {x| x ? R and x ? b}.


19. Soru

What is an absolute value?

Cevap

For a given real number a on the real line, the distance from a to the origin is called the absolute value of a and is denoted by |a|. The absolute value has the following properties:
|a| ? 0, |a + b| ? |a| + |b|, |a . b| = |a| .|b|, |a / b| = (|a|) / (|b|) (b?0) .
Open and closed finite intervals can be represented by using the absolute value.


20. Soru

Represent the intervals [-1, 5] and (4, 6) by using the absolute value.

Cevap

The middle points are c = (-1+ 5)/2 = 2 and c = (4 + 6)/2 = 5 the lengths are b – a = 5 – (-1) = 6 and b – a = 6 – 4 = 2, respectively.

[-1, 5] = {x | x ? R , |x – 2| ? 3},
(4, 6) = {x| x ? R |x – 5| < 1}


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