MATEMATİK I

A set can also be defined by giving a rule that determines whether an element is a member or not. For example, the expression

B = {x: x is a natural number between 10 and 20}

means that x is the collection of all natural numbers greater than 10 and less than 20. The set B above equivalently can be written as

B = {x| x is a natural number between 10 and 20}

or

B = {11, 12, ..., 19}.

Obviously, 15 ∈ B, 10 ∉ B, 25 ∉ B. The notations {x: …} and {x| …} are called implicit representations of sets.

As also understood from the information given, “A = {x| x is a natural number between 30 and 40" and “ A = {x: x is a natural number between 30 and 40}” are the implicit representations of the set  A= {31, 32, 33,…39}, so the correct answer is A.

The expression “A = {x| x is a natural number}” is not well-defined, so it is not a set.

The expression “A is its own subset.” in the option IV explains the notion of set, but it is not the implicit representation of the set A.

Operations on sets are somewhat similar to operations of addition, multiplication and subtraction of numbers.

Let A and B be two sets.

The set of elements that are in either A or B or both is called the union of the sets A and B and is denoted by A ∪ B, i.e.,

A ∪ B = {x| x ∈ A or x ∈ B}

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.

Example

A = {1, 3, 5, 8}, B = {1, 3, 7}. Then A ∪ B = {1, 3, 5, 7, 8}.

If an element appears in both sets then we only list it once in the new set.

Example

A = {x| x is a city in Turkey with population greater than 1 million}, B = {x| x is a city in Turkey with population less than 500 000}. Then A ∪ B ={x| x is a city in Turkey with population greater than 1 million or less than 500 000}. 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 \ B = {x| x ∈ A and x ∉ B}

Example

A = {3, 5, 8, 10}, B = {4, 5, 9}. Then A \ B = {3, 8, 10}.

Example

A = {0, 1, 2, 3, 4, …}, B = {1, 3, 5, 7, …}. Then A \ B = {0, 2, 4, 6, ...}.

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 A^c . That is,

A^c = U \ A = {x| x ∈ U and x ∉ A}

Example

U = {1, 2, …, 10}, A = {9, 10}. Then U \ A = {1, 2, …, 8}.

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}

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

A ∩ B = {2, 3}. x ∈ A ∩ B if and only if x ∈ A and x ∈ B.

Example

A = {x| x is a city in Turkey with population less than 1 million},

B = {x| x is a city in Turkey with population greater than 500 000}.

Then A ∩ B ={x| x is a city in Turkey with population between 500 000 and 1 million}.

As also understood from the information given,

If A = {1, 2, 3, 5, 8} and B = {2, 3, 10, 11}, the expressions in the options;

I “A ∩ B = {2, 3}”

III “A \ B = {1, 5, 8}”

IV “B \ A = {10, 11}” are correct, so the correct answer is D.

If an element appears in both sets then we only list it once in the new set, so the expression in the option II “A ∪ B = {1, 2, 2, 3, 3, 5, 8, 10, 11}” is not correct. It is written as;

A ∪ B = {1, 2, 3, 5, 8, 10, 11}.

 

The statements regarding to sets in the options;

I “The sets A = {4, 8, 11, 15} and B = {8, 15, 4 ,11} are the same.”

II “The empty set Ø is a subset of any set.”

III “The equality A = B is equivalent to two inclusions: A ⊂ B and B ⊂ A.”

IV “The set of elements that are in either A or B or both is called the union of the sets A and B and is denoted by A ∪ B.”

V “x ∈ A ∩ B if and only if x ∈ A and x ∈ B.”

VI “If A ∩ B = Ø then the sets A and B are called disjoint sets.” are correct, so the correct answer is D.

The statement in the option VII “Universal set is unique.” is not correct. Universal set is not unique. It varies depending on the problem.

If A is a finite set and n is the number of its elements, that is s(A) = n, then the number of all subsets of A is 2ⁿ.

For example, if A = {1, 2, 3} then the subsets of A

are Ø, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.

The set A has 2³ = 8 subsets. Recall that, by convention, every set is its own subset, that is A ⊂ A. The empty set Ø is a subset of any set.

As also understood from the information given the subsets of A are

Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, so the correct answer is E.

Example

For given sets A and B assume that s(A) = 10, s(B) = 16 and s(A ∪ B) = 23. Find s(B \ A).

Solution

s(A ∩ B) = s(A) + s(B) – s(A ∪ B) = 10 + 16 – 23 = 3. Therefore s(A \ B) = 7 and  s(B \ A) = 13

As also understood form the solution given for the example, all of the statements in the options are correct, so the correct answer is E.

s(A ∩ B) = s(A) + s(B) – s(A ∪ B) = 20 + 26 – 33 = 13. Therefore s(A \ B) = 7 and  s(B \ A) = 13.

As also understood from the information given, the correct answer is D. The statements regarding to the number sets and the real line in the options I “Every natural number is an integer. Every integer is a rational number.”, II “Every rational number has an infinite number of representations by fractions.” and IV “There is one-to-one correspondence between the points of the real line and the set of real numbers.”are correct. The statement in the option III “Every rational number has a finite periodical representation.” is not correct because of the fact that every rational number has a finite or infinite periodical representation.

Example

600 men are interviewed about which sports they like. It is found that 190 like football, 140 like basketball, 120 like volleyball, 65 like football and basketball, 50 like football and volleyball, 40 like basketball and volleyball, 30 like all three sports.

Solution

Define

F = The set of all men who like football,

B = The set of all men who like basketball,

V = The set of all men who like volleyball.

Then s(F) = 190,    s(B) = 140,   s(V) = 120, s(F ∩ B) = 65,   s(F ∩ V) = 50,   s(B ∩ V) = 40,

s(F ∩ B ∩ V) = 30 where, as defined above, the symbol s(.) is used for the number of elements. Below, we give the Venn diagram solution of this problem where the numbers on the regions they are sitting indicate the number of elements of the corresponding set.

For  example, the number 30 indicates that he number of elements of the intersection F ∩ B ∩ V is 30.

We can easily count the required numbers in the questions:

As also undersood from the information given, all statements in the options are correct according to the case given, so the correct answer is E.

If all the sets under examination are subsets of some set U then U is called a universal set. Given two sets A and B:

As also understood from the information given, the correct answer is D. The statements in the options I “The set of all elements of the sets A and B is called the union of A and B “, II  “The set of the elements that the set A and the set B have in common is called the intersection of A and B.”, and III “The set of all elements of A which are not in B is called the difference of A and B. are correct. The statement in the option IV “The difference of U and A is called the complement of U.” is not correct because of the fact that the difference of U and A is called the complement of A not U.

As also understood from the information given, the correct answer is C.

The statements in the options

I. a⁰ = 1 for all nonzero numbers a.”,

III. “∞ (infinity) is not a number. It represents an infinitely large quantity.”,

IV. “The whole real line is an interval denoted by (–∞, ∞).” and

V. “Powers are used when we multiply a real number by itself repeatedly.”

are correct.

The statement in the option

II. “If a = 0, a⁰ is called power of a.” is not correct because of the fact that if a = 0, a⁰ is called indeterminate.

Let A = (-1, 3), B = (2, 5) be open intervals. Find A ∪ B, A ∩ B, and A \ B.  

A ∪ B = (–1, 5),      A ∩ B = (2, 3),      A \ B = (-1, 2]

As also understood from the information given, the correct answer is E. All of the statements in the options are correct.