**Sets and Union- Keynotes**

**Sets and Union-
Keynotes-aptitude-questions-answers-basics**

**Sets and Union
Aptitude basics, practice questions, answers and explanations **

**Prepare for companies tests and interviews**

A set can be defined as a collection of
things that are brought together because they obey a certain rule. These
'things' may be anything you like: numbers, people, shapes, cities, bits of
text ..., literally anything. The key fact about the 'rule' they all obey is
that it must be well-defined. In other words, it enables us to say for sure
whether or not a given 'thing' belongs to the collection. If the 'things' we're
talking about are English words, for example, a well-defined rule might be:
'... has 5 or more letters'. A rule which is not well-defined (and therefore
couldn't be used to define a set) might be: '... is hard to spell'

**Elements**

A 'thing' that belongs to a given set is
called an element of that set.

For example: Henry VIII is an element of the set of Kings of England

**Notation**

Curly brackets {...... } are used to
stand for the phrase 'the set of ...'. These braces can be used in various
ways.

For example: We may list the elements of
a set: { − 3, − 2, − 1,0,1,2,3}.

We may describe the elements of a set: {
integers between − 3 and 3 inclusive}.

We may use an identifier (the letter x
for example) to represent a typical element, a | symbol to stand for the phrase
'such that', and then the rule or rules that the identifier must obey:

{x | x is an integer and | x | < 4}
or {x|x ϵ Z, |x| <4 }

The last way of writing a set - called
set comprehension notation - can be generalized as:

x | P(x), where P(x) is a statement
(technically a propositional function) about x and the set is the collection of
all elements x for which P is true.

The symbol ϵ is used
as follows:

ϵ means 'is an element of ...'.
For example: dog ϵ {quadrupeds}

ɇ means 'is not an element of ...'.
For example:

Washigton DC ɇ {European
capital cities}

A set can be finite: {British citizens}or
infinite: {7, 14, 21, 28, 35, ?. }.

Sets will usually be denoted using upper
case letters: A, B, ...

Elements will usually be denoted using
lower case letters: x, y, ...

**Some Special Sets**

**1.Universal Set**

The set of all the 'things' currently
under discussion is called the universal set (or sometimes, simply the
universe). It is denoted by U. The universal set doesn?t contain everything in
the whole universe. On the contrary, it restricts us to just those things that
are relevant at a particular time. For example, if in a given situation we?re
talking about numeric values ? quantities, sizes, times, weights, or whatever ?
the universal set will be a suitable set of numbers (see below). In another
context, the universal set may be {alphabetic characters} or {all living
people}, etc.

**2.Empty set**

The set containing no elements at all is
called the null set, or empty set. It is denoted by a pair of empty braces: { }
or by the symbol f. It may seem odd to define a set that contains no elements.
Bear in mind, however, that one may be looking for solutions to a problem where
it isn't clear at the outset whether or not such solutions even exist. If it
turns out that there isn't a solution, then the set of solutions is empty.

For example:

If U = {words in the English language}
then {words with more than 50 letters}= f .

If U = {whole numbers} then {x|x2 = 10}
= f .

Operations on the empty set

Operations performed on the empty set
(as a set of things to be operated upon) can also be confusing. (Such
operations are nullary operations.) For example, the sum of the elements of the
empty set is zero, but the product of the elements of the empty set is one (see
empty product). This may seem odd, since there are no elements of the empty
set, so how could it matter whether they are added or multiplied (since ?they?
do not exist)? Ultimately, the results of these operations say more about the
operation in question than about the empty set. For instance, notice that zero
is the identity element for addition, and one is the identity element for
multiplication.

**3.Equality**

Two sets A and B are said to be equal if
and only if they have exactly the same elements. In this case, we simply write:

A = B

Note two further facts about equal sets:

The order in which elements are listed
does not matter.

If an element is listed more than once,
any repeat occurrences are ignored.

So, for example, the following sets are
all equal:

{1, 2, 3} = {3, 2, 1} = {1, 1, 2, 3, 2,
2}

(You may wonder why one would ever come
to write a set like {1, 1, 2, 3, 2, 2}. You may recall that when we defined the
empty set we noted that there may be no solutions to a particular problem -
hence the need for an empty set. Well, here we may be trying several different
approaches to solving a problem, some of which in fact lead us to the same
solution. When we come to consider the distinct solutions, however, any such
repetitions would be ignored.)

**4.Subsets**

If all the elements of a set A are also
elements of a set B, then we say that A is a subset of B, and we write: A ⊆ B

For example: If T = {2, 4, 6, 8,
10} and E = {even integers}, then T ⊆ E

If A = {alphanumeric characters} and P =
{printable characters}, then A ⊆ P

If Q = {quadrilaterals} and F = {plane
figures bounded by four straight lines}, then Q ⊆ F

Notice that A ⊆ B does not
imply that B must necessarily contain extra elements that are not in A; the two
sets could be equal ? as indeed Q and F are above. However, if, in addition, B
does contain at least one element that isn?t in A, then we say that A is a
proper subset of B. In such a case we would write: A ⊂ B

In the examples above:

E contains 12, 14, ... , so T ⊂ E

P contains $, ;, &, ..., so A ⊂ P

But Q and F are just different ways of
saying the same thing, so Q = F.

The use of ⊂ and ⊆ is clearly
analogous to the use of < and ≤ when comparing two numbers.

Note: Every set is a subset of the
universal set, and the empty set is a subset of every set.

**5.Disjoint**

Two sets are said to be disjoint if they
have no elements in common.

For example: If A = {even numbers} and B
= {1, 3, 5, 11, 19}, then A and B are disjoint.

**Operations on Sets**

**1.Intersection**

The intersection of two sets A and B,
written A ∩ B, is the set of elements that are in A and in B.

(Note that in symbolic logic, a similar
symbol,^, is used to connect two logical propositions with the AND operator.)

For example, if A = {1, 2, 3, 4} and B =
{2, 4, 6, 8}, then A ∩ B = {2, 4}.

We can say, then, that we have combined
two sets to form a third set using the operation of intersection.

**2.Union**

In a similar way we can define the union
of two sets as follows:

The union of two sets A and B, written A
∪ B, is the set
of elements that are in A or in B (or both).

(Again, in logic a similar symbol,V, is
used to connect two propositions with the OR operator.)

So, for example, {1, 2, 3, 4} ∪ {2, 4, 6, 8} =
{1, 2, 3, 4, 6, 8}.

You'll see, then, that in order to get
into the intersection, an element must answer 'Yes' to both questions, whereas
to get into the union, either answer may be 'Yes'.

The ∪ symbol looks like the first letter of
'Union' and like a cup that will hold a lot of items. The ∩ symbol looks
like a spilled cup that won't hold a lot of items, or possibly the letter 'n',
for intersection. Take care not to confuse the two.

**3.Difference**

The difference of two sets A and B (also
known as the set-theoretic difference of A and B, or the relative complement of
B in A) is the set of elements that are in A but not in B.

This is written A - B, or sometimes A \
B.

For example, if A = {1, 2, 3, 4} and B =
{2, 4, 6, 8}, then A - B = {1, 3}.

**4.Complement**

The set of elements that are not in a
set A is called the complement of A. It is written A′ (or sometimes AC,
or ?). Clearly, this is the set of elements that answer 'No' to the question
Are you in A?.

For example, if U = N and A = {odd
numbers}, then A′ = {even numbers}.

Notice the spelling of the word
complement: its literal meaning is 'a complementary item or items'; in other
words, 'that which completes'. So if we already have the elements of A, the
complement of A is the set that completes the universal set.

**5.Cardinality**

The cardinality of a finite set A,
written | A | (sometimes #(A) or n(A)), is the number of (distinct) elements in
A. So, for example:

If A = {lower case letters of the
alphabet}, | A | = 26.

**Some special sets of numbers**

Several sets are used so often, they are
given special symbols.

**1.The natural numbers**

The 'counting' numbers (or whole
numbers) starting at 1, are called the natural numbers. This set is sometimes
denoted by N. So N = {0, 1, 2, 3, ...}.

Note that, when we write this set by
hand, we can't write in bold type so we write an N in blackboard bold font: N

**2.Integers**

All whole numbers, positive, negative
and zero form the set of integers. It is sometimes denoted by Z. So Z = {...,
-3, -2, -1, 0, 1, 2, 3, ...}

In blackboard bold, it looks like this:
Z

**3.Real numbers**

If we expand the set of integers to
include all decimal numbers, we form the set of real numbers. The set of reals
is sometimes denoted by R.

A real number may have a finite number
of digits after the decimal point (e.g. 3.625), or an infinite number of
decimal digits. In the case of an infinite number of digits, these digits may:

recur; e.g. __8.127127127__...

... or they may not recur; e.g.
3.141592653...

In blackboard bold: R

**4.Rational numbers**

Those real numbers whose decimal digits
are finite in number, or which recur, are called rational numbers. The set of
rationals is sometimes denoted by the letter Q.

A rational number can always be written
as exact fraction p/q; where p and q are integers. If q equals 1, the fraction
is just the integer p. Note that q may NOT equal zero as the value is then
undefined.

For example: 0.5, -17, 2/17, 82.01,
3.282828... are all rational numbers.

In blackboard bold: Q

**5.Irrational numbers**

If a number can't be represented exactly
by a fraction p/q, it is said to be irrational.

Examples include: √2, √3.