Learn CBSE & NCERT Solutions for Class 11 : Topic SETS

NCERT Solutions for Class 11 : Topic SETS

CBSE Class 11 Mathematics : A set is any un ordered collection of distinct objects. These objects are called the elements or members of the set. The set containing no elements is known as the empty set. A set may have finitely many elements, such as the set of desks in a classroom; or infinitely many members, such as the set of positive integers; or possibly no elements at all. The members of a set can be practically any objects imaginable, as long as they are clearly defined. Thus a set might contain numbers, letters polynomials, points, colors, or even other sets. In theory, a set could contain any combination of these objects, but in practice, we tend to only consider sets whose elements are related to one another in some way, such as the set of letters in your name, or the set of even numbers. We typically name sets using upper case letters, such as A, B or C. There are a variety of ways to describe the elements of a set, each of which has advantages.

We could give a verbal description of a set, for example, by declaring that B is the set of letters in the title of this book. We might also simply list the elements of a set within curly brackets:

B = {b, r, i, d, g, e, t, o, h, i, g, h, e, r, m, a, t, h}.

Since the order in which elements are listed is irrelevant, we could also write

B = {a, b, d, e, g, h, i, m, o, r, t} or B = {m, o, t, h, b, r, i, g, a, d, e}.

For a given set, it is natural to ask which objects are included in the set and how many objects there are in total. We indicate membership in or exclusion from a set using the symbols ∈ and “∈. Thus it would be fair to say that a ∈ Band g ∈ B, but z “∈ B and ! “∈ B either. We also write |B| to indicate the size, or cardinality of set B. In the example above we have |B| = 11, of course. For the time being, we will only consider the cardinality of finite sets.

INTERSECTION AND UNION:

The set of elements common to two given sets A and B is known as the intersection and written as A ∩ B. The set of elements appearing in at least one of these sets is called the union, denoted by A ∪ B.

EXAMPLE: Decide which elements ought to belong to each of A ∪ B ∪ C and A ∩ B ∩ C. Then write a compact description of each set using bar notation.

Note that the set operation of intersection corresponds to the logical operation of conjunction. This relationship is made clear by the fact that

A ∩ B = {x | x ∈ A and x ∈ B}.

Similarly, union corresponds to the logical operation of disjunction, since

A ∪ B = {x | x ∈ A or x ∈ B}.

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We say that A and B are equal sets, written A = B, if these two sets contain precisely the same elements. One common technique for showing that two sets are equal is to show that every element of the first set must be an element of the second set, and vice-versa.

To establish the set identity A ∩ B = A ∪ B we use these two strategies.

Step one: Let x be any element of the first set; i.e. let x ∈A ∩ B. This means that x “∈ A ∩ B. Since A ∩ B consists of elements in both A and B, if x is not in the intersection then either x “∈ A or x “∈ B, or both. In other words, x ∈ Aor x ∈ B, which means that x ∈ A ∪ B.

Step two: On the other hand, if x ∈ A ∪ B then we know that x ∈ A or x ∈ B, which means that x “∈ A or x “∈ B. Since x is missing from at least one of the sets A or B, it cannot reside in their intersection, hence x “∈ A∩B. Finally, this is the same as x ∈ A ∩ B. Hence, we conclude that the sets A ∩ B and A ∪ Bare indeed equal.

VENN DIAGRAM:

A Venn diagram uses overlapping circles or other shapes to illustrate the logical relationships between two or more sets of items. Given below Venn diagram showing A ∩ B and A ∪ B.

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SUB SETS AND POWER SETS:

Given sets A and B, whenever each element of A is also an element of Bwe say that A is a subset of B and write A ⊆ B. Therefore, to prove that A ⊆ B one must show that if x ∈ A, then x ∈ B.

On the other hand, if A and B have no elements in common than they are disjoint, which can be proved by showing that if x ∈ A then x “∈ B.It makes sense that if A is a subset of B, then B contains A. More

formally, we say that B is a superset of A, denoted by B ⊇ A. However, this perspective(and associated notation) arises fairly infrequently.

We define the power set P(A) of a set A to be the set of all subsets of A, including the empty set and the set A itself.

INDEX SETS:

The subscripts a, b, . . . are known as indices; the set I = {a, b, c,…, z}of all indices is called the index set. The collection of all the sets W_athrough W_zcomprises a family of sets, in the sense that they are related by a common definition. It may help to remember that each index indicates a particular set in the family.

Our definition of the intersection of the family of sets Wα can be shortened to”

П α∈IWα = {x| x ∈ Wα for all α∈ I}.

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