Operations on Sets

Now that we have an idea about what a set is, we can move on to understand how to work with them. Is there a way to merge two sets together? What is the intersection of sets and how do we find the intersection of two sets? Should we even be allowed to add two sets together? Here we will find out all the answers!

Operations on Sets

Operations on Sets

Unlike the real world operations, mathematical operations do not require a separate no-contamination room, surgical gloves, and masks. But certainly, expertise to solve the problem, special tools, techniques, and tricks as well as knowledge of all the basic concepts are required to obtain a solution. Following are some of the operations that are performed on the sets: –

Union
Intersection
Difference
Complement
Let’s deal with them one by one.

Union of Sets
Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12}. Then, A U B is represented as the set containing all the elements that belong to both the sets individually. Mathematically,

A U B = {x : x ϵ A or x ϵ B}

So, A U B = {2, 4, 6, 8, 10, 12},

here the common elements are not repeated.

Properties of (A U B)
Commutative law holds true as (A U B) = (B U A)
Associative law also holds true as (A U B) U {C} = {A} U (B U C)
Let A = {1, 2} B = {3, 4} and C = {5, 6}
A U B = {1, 2, 3, 4} and (A U B) U C = {1, 2, 3, 4, 5, 6}
B U C = {3, 4, 5, 6} and A U (B U C) = {1, 2, 3, 4, 5, 6}
Thus, the law holds true and is verified.

A U φ = A (Law of identity element)
Idempotent Law – A U A = A
Law of the Universal set (U): (A U U) = U


Intersection of Sets
An intersection is the collection of all the elements that are common to all the sets under consideration. Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12} then A ∩ B or “A intersection B” is given by:

“A intersection B” or A ∩ B = {6, 8}

Mathematically, A ∩ B = {x : x ϵ A and x ϵ B}

Properties of the Intersection – A ∩ B
The intersection of the sets has the following properties:

Commutative law – A ∩ B = B∩ A
Associative law – (A ∩ B)∩ C = A ∩ (B∩ C)
φ ∩ A = φ
U ∩ A = A
A∩ A = A; Idempotent law.
Distributive law – A ∩ (BU C) = (A ∩ B) U (A ∩ C)
Difference of Sets
The difference of set A and B is represented as:

A – B = {x : x ϵ A and x ϵ B}

Conversely, B – A = {x : x ϵ A and x ϵ B}

Let, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then A – B = {1, 3, 5} and B – A = {8}. The sets (A – B), (B – A) and (A ∩ B) are mutually disjoint sets; it means that there is NO element common to any of the three sets and the intersection of any of the two or all the three sets will result in a null or void or empty set.


Complement of Sets
If U represents the Universal set and any set A is the subset of A then the complement of set A (represented as A’) will contain ALL the elements which belong to the Universal set U but NOT to set A.

Mathematically, A’ = U – A

Alternatively, the complement of a set A, A’ is the difference between the universal set U and the set A.

Properties of Complement Sets
A U A’ = U
A ∩ A’ = φ
De Morgan’s Law – (A U B)’ = A’ ∩ B’ OR (A ∩ B)’ = A’ U B’
Law of double complementation : (A’)’ = A
φ’ = U
U’ = φ
Hence, these are the basic concepts and operations on Sets.



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https://www.toppr.com/guides/maths/sets/operations-on-sets/

INTEGER

An integer (pronounced IN-tuh-jer) is a whole number (not a fractional number) that can be positive, negative, or zero.

INTEGER

Examples of integers are: -5, 1, 5, 8, 97, and 3,043.

Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14, .09, and 5,643.1.

The set of integers, denoted Z, is formally defined as follows:

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

In mathematical equations, unknown or unspecified integers are represented by lowercase, italicized letters from the "late middle" of the alphabet. The most common are p, q, r, and s.

The set Z is a denumerable set. Denumerability refers to the fact that, even though there might be an infinite number of elements in a set, those elements can be denoted by a list that implies the identity of every element in the set. For example, it is intuitive from the list {..., -3, -2, -1, 0, 1, 2, 3, ...} that 356,804,251 and -67,332 are integers, but 356,804,251.5, -67,332.89, -4/3, and 0.232323 ... are not.

The elements of Z can be paired off one-to-one with the elements of N, the set of natural numbers, with no elements being left out of either set. Let N = {1, 2, 3, ...}.

In infinite sets, the existence of a one-to-one correspondence is the litmus test for determining cardinality, or size. The set of natural numbers and the set of rational numbers have the same cardinality as Z. However, the sets of real numbers, imaginary numbers, and complex numbers have cardinality larger than that of Z.





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https://whatis.techtarget.com/definition/integer

Operations of Arithmetic

The first thing to note is that in algebra we use letters as well as numbers. But the letters represent numbers. We imitate the rules of arithmetic with letters, because we mean that the rule will be true for any numbers.

Here, for example, is the rule for adding fractions:

a/c + b/c = a/c + b/c

The letters a and b mean: The numbers that are in the numerators. The letter c means: The number in the denominator. The rule means:

"Whatever those numbers are, add the numerators
and write their sum over the common denominator."

Algebra is telling us how to do any problem that looks like that. That is one reason why we use letters.

(The symbols for numbers, after all, are nothing but written marks. And so are lettersexclamation As the student will see, algebra depends only on the patterns that the symbols make.)

The numbers are the numerical symbols, while the letters are called literal symbols.

The four operations of arithmetic

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Do the problem yourself first!

1. Addition:  a + b.   The operation sign is + , and is called the plus sign.  Read a + b as "a plus b. For example, if a represents 3, and b represents 4, then a + b represents 7.
2. Subtraction:  a − b.   The operation sign is − , and is called the minus sign.  Read a − b as "a minus b." If a represents 8, for example, and b represents 2, then a − b represents 6.
3. Multiplication:  a· b.  Read a· b as "a times b." The multiplication sign in algebra is a centered dot.  We do not use the multiplication cross ×, because we do not want to confuse it with the letter x. And so if a represents 2, and b represents 5, then a· b = 2· 5 = 10. "2 times 5 equals 10." Do not confuse the centered dot -- 2·5, which in the United States means multiplication -- with the decimal point:  2.5. However, we often omit the multiplication dot and simply write ab.  Read "a, b."  In other words, when there is no operation sign between two letters, or between a letter and a number, it always means multiplication.  2x  means  2 times x.
4. Division:  a/b. Read as "a divided by b." In algebra, we use the horizontal division bar.  If a represents 10, for example and b represents 2, then a/b  =   10/ 2   =  5. "10 divided by 2 is 5."

Note:  In algebra we call  a + b  a "sum" even though we do not name an answer.  As the student will see, we name something in algebra simply by how it looks.  In fact, you will see that you do algebra with your eyes, and then what you write on the paper, follows.

Similarly, we call  a − b  a difference,  ab

 a product, and  a
b  a quotient.
This sign = of course is the equal sign, and we read this --

a = b

-- as "a equals (or is equal to) b."

That means that the number on the left that a represents, is equal to the number on the right that b represents.  If we write

a + b = c,

and if a represents 5, and b represents 6, then c must represent 11.



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Source:

https://www.themathpage.com/alg/algebraic-expressions.htm#parentheses

Properties of Integers

We will learn about properties of integers. This is resume of integers properti

properties of integers

Let us now study these properties in detail.

Closure Property

The System of Integers in Addition
It states that addition of two Integers always results in an Integer. For example, 7 + 4 = 11,  the result we get is an integer. Therefore, the system is closed under addition.

The System of Integers under Subtraction
It states that subtraction of two Integers always results in an Integer. For example, 7 – 4 = 3, the result we get is an integer. Also, 2 – 4 = -2. The result is also an integer. Therefore, the system is closed under subtraction.

The System of Integers under Multiplication
It states that multiplication of two integers always results in an integer. For example, 7 × 4 = 28, the result we get is an integer. Therefore, the system is closed under multiplication.

The System of Integers under Division
It states that division of two integers does not always result in an integer. For example, 7 ÷ 4 = 74, the result we get is not an integer. But, 8 ÷ 4 = 2, the result we get is an integer. Therefore, a system is not closed under division.

Commutative Property

It is a property that associates with binary operations or functions like addition, multiplication. Take any two numbers a and b and subtract them. That is a – b, say 5 – (-3). Now subtract a from b. That is b – a or -3 – 5. Are they same? No, they are not equal. So, the commutative property does not hold for subtraction. Similarly, it does not hold for division too.

Again take any two numbers a and b and add them. That is a + b. Now add b and a which comes to be b+ a. Aren’t the same? Yes, they are equal because of commutative property which says that we can swap the numbers and still we get the same answer.

Associative Property

Associative property of integers states that for any three elements(numbers) a, b and c

1) For Addition a + ( b + c ) = ( a + b ) + c
2) For Multiplication a × ( b × c ) = ( a × b ) × c
3) For Subtraction. Associative property does not hold for subtraction a – ( b – c ) != ( a – b ) – c
4) For Division. Associative property does not hold for division a ÷ ( b ÷ c ) != ( a ÷ b ) ÷ c

Multiplicative Identity for Integers
The multiplicative identity of any integer a  is a number b which when multiplied with a, leaves it unchanged, i.e. b is called as the multiplicative identity of any integer a if a× b = a. Now, when we multiply 1 with any of the integers a we get a × 1 = a = 1 × a  So, 1 is the multiplicative identity for integers.

Additive Identity for Integers

The additive identity of any integer a  is a number b which when added with a, leaves it unchanged, i.e. b is called as the additive identity of any integer a if a + b = a. Now, when we add 0 with any of the integers a we get a + 0 = a = 0 + a  So, 0 is the additive identity for integers.

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https://www.toppr.com/guides/maths/rational-numbers/properties-of-integers/

Square Number

When you multiply a whole number (not a fraction) by itself, the result is a square number. For example 3 x 3 = 9. Nine is the square of three multiplied by itself.
As mathematicians often try to shorten the way things are written, we show that we want to square a number by simply writing that number with a small  '2' to the right of it.

Source: 34auburn.weebly.com

Squaring Negative Numbers

As you may know already, if you multiply a negative number by another negative number, it becomes a positive.

Example:  -3 x -3 would become 9 just the same as it would if both the 3’s were positive!

However, if you are multiplying a negative with a positive, like -3 x 3 it would become negative -9 and then, of course, it wouldn’t be a square number (because -3 is a different number to 3)!

Squaring Decimals

Just like whole numbers (integers), it’s easy to square a decimal number too!

1.23 Squared = 1.23 × 1.23 = 1.5129

Square Root

A square root is a number that’s been multiplied to get the square number. For example the square root of 9 is 3 because 3 x 3 = 9.

Finding the square root of a number is much trickier than calculating the square number in the first place, so many calculators have a square root button. This is the one that looks like a tick √. It’s called the radical.


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https://www.edplace.com/blog/what-are-square-numbers-square-numbers-explained/

Finding General Term of Squence

A sequence is a function whose domain is an ordered list of numbers. These numbers are positive integers starting with 1. Sometimes, people mistakenly use the terms series and sequence. A sequence is a set of positive integers while series is the sum of these positive integers. The denotation for the terms in a sequence is:

a1, a2, a3, a4, an, . . .

Finding the nth term of a sequence is easy given a general equation. But doing it the other way around is a struggle. Finding a general equation for a given sequence requires a lot of thinking and practice but, learning the specific rule guides you in discovering the general equation. In this article, you will learn how to induce the patterns of sequences and write the general term when given the first few terms. There is a step-by-step guide for you to follow and understand the process and provide you with clear and correct computations.

An arithmetic series is a series of ordered numbers with a constant difference. In an arithmetic sequence, you will observe that each pair of consecutive terms differs by the same amount. For example, here are the first five terms of the series.

3, 8, 13, 18, 23

Do you notice a special pattern? It is obvious that each number after the first is five more than the preceding term. Meaning, the common difference of the sequence is five. Usually, the formula for the nth term of an arithmetic sequence whose first term is a1 and whose common difference is d is displayed below.

an = a1 + (n - 1) d

Source: teacherspayteachers.com

Steps in Finding the General Formula of Arithmetic and Geometric Sequences
1. Create a table with headings n and an where n denotes the set of consecutive positive integers, and an represents the term corresponding to the positive integers. You may pick only the first five terms of the sequence. For example, tabulate the series 5, 10, 15, 20, 25, . . .

2. Solve the first common difference of a. Consider the solution as a tree diagram. There are two conditions for this step. This process applies only to sequences whose nature are either linear or quadratic.

Condition 1: If the first common difference is a constant, use the linear equation ax + b = 0 in finding the general term of the sequence.

a. Pick two pairs of numbers from the table and form two equations. The value of n from the table corresponds to the x in the linear equation, and the value of an corresponds to the 0 in the linear equation.

b. After forming the two equations, calculate a and b using the subtraction method.

c. Substitute a and b to the general term.

d. Check if the general term is correct by substituting the values in the general equation. If the general term does not meet the sequence, there is an error with your calculations.

Condition 2: If the first difference is not constant and the second difference is constant, use the quadratic equation

a. Pick three pairs of numbers from the table and form three equations. The value of n from the table corresponds to the x in the linear equation, and the value of an corresponds to the 0 in the linear equation.


b. After forming the three equations, calculate a, b, and c using the subtraction method.

c. Substitute a, b, and c to the general term.

d. Check if the general term is correct by substituting the values in the general equation. If the general term does not meet the sequence, there is an error with your calculations.




This article was originally published on The owlcation. Read the original article.


Source:
https://owlcation.com/stem/How-to-Find-the-General-Term-of-Arithmetic-and-Geometric-Sequences

Definition of Set

A set is a collection of objects, things or symbols which are clearly defined.
The individual objects in a set are called the members or elements of the set.

The following table shows some Set Theory Symbols.

A set must be properly defined so that we can find out whether an object is a member of the set.

1. Listing the elements (Roster Method)
The set can be defined by listing all its elements, separated by commas and enclosed within braces. This is called the roster method.

Example:
B = {2, 4, 6, 8, 10}
X = {a, b, c, d, e}

However, in some instances, it may not be possible to list all the elements of a set. In such cases, we could define the set by method 2.

2. Describing the elements
The set can be defined, where possible, by describing the elements. This is called the set-builder notation.

Example:
C = {x : x is an integer, x > – 3 }
This is read as: “C is the set of elements x such that x is an integer greater than –3.”

D= {x: x is a river in a state}

We should describe a certain property which all the elements x, in a set, have in common so that we can know whether a particular thing belongs to the set.

We relate a member and a set using the symbol ∈. If an object x is an element of set A, we write x ∈ A. If an object z is not an element of set A, we write z ∉ A.

∈ denotes “is an element of’ or “is a member of” or “belongs to”

∉ denotes “is not an element of” or “is not a member of” or “does not belong to”

Example:
If A = {1, 3, 5} then 1 ∈ A and 2 ∉ A

Basic vocabulary used in set theory

A set is a collection of distinct objects. The objects can be called elements or members of the set.
A set does not list an element more than once since an element is either a member of the set or it is not.

There are three main ways to identify a set:
1. A written description
2. List or Roster method
3. Set builder Notation

The empty set or null set is the set that has no elements.
The cardinality or cardinal number of a set is the number of elements in a set.
Two sets are equivalent if they contain the same number of elements.
Two sets are equal if they contain the exact same elements although their order can be different.

Definition and notation used for subsets and proper subsets

If every member of set A is also a member of set B, then A is a subset of B, we write A ⊆ B. We can also say A is contained in B.

If A is a subset of B, but A is not equal B then A is a proper subset of B, we write A ⊂ B.

The empty set is a subset of any set.
If a set A has n elements that it has 2n subsets.

How to use Venn diagrams to show relationship between sets and set operations?

A Venn diagram is a visual diagram that shows the relationship of sets with one another.
The set of all elements being considered is called the universal set (U) and is represented by a rectangle.

The complement of A, A', is the set of elements in U that is not in A.
Sets are disjoint if they do not share any elements.
The intersection of A and B is the set of elements in both set A and set B.
The union of A and B is the set of elements in either set A or set B or both.



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Source:
https://www.onlinemathlearning.com/describing-sets.html