# sets

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Category: Education

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### Slide 1:

DONE BY, ashams k xi b k.v pattom

sets

### iNTRODUCTION :

In mathematics, collection of entities, called elements of the set, that may be real objects or conceptual entities. iNTRODUCTION

### Slide 4:

The intersection of two sets is made up of the objects contained in both sets, shown in a Venn diagram.

### Slide 5:

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics, Education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

### Slide 6:

In philosophy, sets are ordinarily considered to be abstract objects[1][2][3][4] physically represented by groups of objects. For instance; three cups on a table when spoken of together as "the cups", or the chalk lines on a board in the form of the opening and closing curly bracket symbols along with any other symbols in between the two bracket symbols. However, proponents of mathematical realism including Penelope Maddy have argued that sets are concrete objects.

### DEFINITION :

By a "set" we mean any collection M into a whole of definite, distinct objects m (which are called the "elements" of M) of our perception [Anschauung] or of our thought. The elements of a set, also called its members, can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. The statement that sets A and B are equal means that they have precisely the same members (i.e., every member of A is also a member of B and vice versa). DEFINITION

SET A AND B

### Slide 9:

Unlike a multiset, every element of a set must be unique; no two members may be identical. All set operations preserve the property that each element of the set is unique. The order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple. As discussed below, in formal mathematics the definition given above turned out to be inadequate; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms. The most basic properties are that a set "has" elements, and that two sets are equal (one and the same) if they have the same elements.

### DESCRIBING SET :

There are two ways of describing, or specifying the members of, a set. One way is by intentional definition, using a rule or semantic description. See this example: A is the set whose members are the first four positive integers. B is the set of colors of the French flag. The second way is by extension, that is, listing each member of the set. An extensional definition is notated by enclosing the list of members in braces: . DESCRIBING SET

### Slide 11:

C = {4, 2, 1, 3} D = {blue, white, red} The order in which the elements of a set are listed in an extensional definition is irrelevant, as are any repetitions in the list. For example, {6, 11} = {11, 6} = {11, 11, 6, 11} are equivalent, because the extensional specification means merely that each of the elements listed is a member of the set. For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive whole numbers may be specified extensionally as:

### Slide 12:

{1, 2, 3, ..., 1000}, where the ellipsis ("...") indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members. Thus the set of positive even numbers can be written as {2, 4, 6, 8, ... }. The notation with braces may also be used in an in tensional specification of a set. In this usage, the braces have the meaning "the set of all ..." So E = {playing-card suits} is the set

### Slide 13:

whose four members are ?, ?, ?, and ?. A more general form of this is set-builder notation, through which, for instance, the set F of the twenty smallest integers that are four less than perfect squares can be denoted: F = {n2 - 4 : n is an integer; and 0 = n = 19} In this notation, the colon (":") means "such that", and the description can be interpreted as "F is the set of all numbers of the form n2 - 4, such that n is a whole number in the range from 0 to 19 inclusive."

### Slide 14:

Sometimes the vertical bar ("|") or the semicolon (";") is used instead of the colon. One often has the choice of specifying a set intensionally or extensionally. In the examples above, for instance, A = C and B = D

A INTERSECTION B

### CARDINALITY :

The cardinality |?S?| of a set S is "the number of members of S." For example, since the French flag has three colors, |?B?| = 3. In mathematical theory, a set {45, 6, 7, 768} has a cardinality value of 4. There is a set with no members and zero cardinality, which is called the empty set (or the null set) and is denoted by the symbol Ø CARDINALITY

### Slide 17:

. For example, the set A of all three-sided squares has zero members ?A? and thus A = Ø. Though it may seem trivial, the empty set, like the number zero, is important in mathematics; indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory. Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others.

### Slide 18:

For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of an entire plane, and indeed of any Euclidean space.

### SUBSET :

If every member of set A is also a member of set B, then A is said to be a subset of B, written A ? B (also pronounced A is contained in B). Equivalently, we can write B ? A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ? is called inclusion or containment. SUBSET

### Slide 21:

If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A ? B (A is a proper subset of B) or B ? A (B is proper superset of A). Note that the expressions A ? B and A ? B are used differently by different authors; some authors use them to mean the same as A ? B (respectively A ? B), whereas other use them to mean the same as A ? B (respectively A ? B).

### Slide 22:

Example: The set of all men is a proper subset of the set of all people. {1, 3} ? {1, 2, 3, 4}. {1, 2, 3, 4} ? {1, 2, 3, 4}. The empty set is a subset of every set and every set is a subset of itself: Ø ? A. A ? A. An obvious but very handy identity, which can often be used to show that two seemingly different sets are equal: A = B if and only if A ? B and B ? A.

### Slide 23:

A is a subset of B

### POWER SET :

The power set of a set S can be defined as the set of all subsets of S. This includes the subsets formed from all the members of S and the empty set. If a finite set S has cardinality n then the power set of S has cardinality 2n. The power set can be written as P(S). If S is an infinite (either countable or uncountable) set then the power set of S is always uncountable. Moreover, if S is a set, then there is never a bijection from S onto P(S). In other words, the power set of S is always strictly "bigger" than S. POWER SET

### Slide 25:

As an example, the power set P({1, 2, 3}) of {1, 2, 3} is equal to the set {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, Ø}. The cardinality of the original set is 3, and the cardinality of the power set is 23, or 8. This relationship is one of the reasons for the terminology power set. Similarly, its notation is an example of a general convention providing notations for sets based on their cardinalities.

### VENN DIAGRAM :

In mathematics, a diagram representing a set or sets and the logical relationships between them. The sets are drawn as circles. An area of overlap between two circles (sets) contains elements that are common to both sets, and thus represents a third set. Circles that do not overlap represent sets with no elements in common (disjoint sets). The method is named after the English logician John Venn. VENN DIAGRAM

### Slide 27:

Sets and their relationships are often represented by Venn diagrams. The sets are drawn as circles the area of overlap between the circles shows elements that are common to each set, and thus represent a third set. (Top) A Venn diagram of two intersecting sets and (bottom) a Venn diagram showing the set of whole numbers from 1 to 20 and the subsets P and O of prime and odd numbers, respectively. The intersection of P and O contains all the prime numbers that are also odd.

### operATIONS ON SETS :

There are three basic set operations: intersection, union, and complementation. operATIONS ON SETS

### 1.UNION :

There are ways to construct new sets from existing ones. Two sets can be "added" together. The union of A and B, denoted by A ? B, is the set of all things which are members of either A or B. Examples: {1, 2} ? {red, white} = {1, 2, red, white}. {1, 2, green} ? {red, white, green} = {1, 2, red, white, green}. {1, 2} ? {1, 2} = {1, 2}. 1.UNION

### Slide 30:

Some basic properties of unions are: A ? B = B ? A. A ? (B ? C) = (A ? B) ? C. A ? (A ? B). A ? A = A. A ? Ø = A. A ? B if and only if A ? B = B.

### Slide 31:

The union of A and B, or A ? B

### 2. INTERSECTION :

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A n B, is the set of all things which are members of both A and B. If A n B = Ø, then A and B are said to be disjoint. Examples: {1, 2} n {red, white} = Ø. {1, 2, green} n {red, white, green} = {green}. {1, 2} n {1, 2} = {1, 2}. 2. INTERSECTION

### Slide 33:

Some basic properties of intersections: A n B = B n A. A n (B n C) = (A n B) n C. A n B ? A. A n A = A. A n Ø = Ø. A ? B if and only if A n B = A.

### Slide 34:

The intersection of A and B, or A n B.

### 3. complement :

Two sets can also be "subtracted". The relative complement of A in B (also called the set theoretic difference of B and A), denoted by B \A, (or B -A) is the set of all elements which are members of B, but not members of A. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect. In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A'. 3. complement

### Slide 36:

Examples: {1, 2} \ {red, white} = {1, 2}. {1, 2, green} \ {red, white, green} = {1, 2}. {1, 2} \ {1, 2} = Ø. {1, 2, 3, 4} \ {1, 3} = {2, 4}. If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then the complement of E in U is O, or equivalently, E' = O.

### Slide 37:

Some basic properties of complements: A ? A' = U. A n A' = Ø. (A')' = A. A \ A = Ø. U' = Ø and Ø' = U. A \ B = A n B'.

### Slide 38:

The relative complementof A in B.

### Slide 39:

The complement of A in U.

### cARTESIAN product :

A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B. Examples: {1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2, white)}. {1, 2, green} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green), (green, red), (green, white), (green, green)}. {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}. cARTESIAN product

### Slide 41:

Some basic properties of cartesian products: A × Ø = Ø. A × (B ? C) = (A × B) ? (A × C). (A ? B) × C = (A × C) ? (B × C). Let A and B be finite sets. Then |?A × B?| = |?B × A?| = |?A?| × |?B?|.

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