SETS, RELATIONS AND FUNCTIONS SUMMARY FOR CA FOUNDATION 2018 EXAMS.

- A set is defined to be a collection of well-defined distinct objects. This collection may be listed or described. Each object is called an element of the set. We usually denote sets by capital letters and their elements by small letters.
- Singleton Set: A set containing one element is called Singleton.
- Equal Set: Two sets A & B are said to be equal, written as A = B if every element of A is in B and every element of B is in A.
- Universal Set: The set which contains all the elements under consideration in a particular problem is called the universal set denoted by S. Suppose that P is a subset of S. Then the complement of P, written as P
^{c }(or P’) contains all the elements in S but not in P. This can also be written as S – P or S ~ P. S – P = {x : x ∈ S, x ∉ P}. - If A and B are two sets then

n(AUB) = n(A) + n(B) – n(A ∩ B)

- If A and B are disjoint sets, then n(AUB) = n(A) + n(B) as n (A ∩ B) = 0
- For three sets P, Q and R

n(PUQUR) = n(P) + n(Q) + n(R) – n(P∩Q) – n(Q R) – n(P∩R) + n(P∩Q∩R)

When P, Q and R are disjoint sets

= n(P) + n(Q) + n(R) - Equivalent Set: Two finite sets A & B are said to be equivalent if n (A) = n(B).
- Power Set: The collection of all possible subsets of a given set A is called the power set of A, to be denoted by P(A).

1. A set containing n elements has 2^{n}subsets.

2. A set containing n elements has 2^{n}-1 proper subsets - Ordered Pair: Two elements a and b, listed in a specific order, form an ordered pair, denoted by (a, b).
- Cartesian Product of sets: If A and B are two non-empty sets, then the set of all ordered pairs (a, b) such that a belongs to A and b belongs to B , is called the Cartesian product of A and B, to be denoted by A × B.

Thus, A × B = {(a, b) : a ∈ A and b ∈ B}

If A = Φ or B = Φ , we define A × B = Φ - Relation and Function: Any subset of the product set X.Y is said to define a relation from X to Y and any relation from X to Y in which no two different ordered pairs have the same first element is called a function.

Let A and B be two non-empty sets. Then, a rule or a correspondence f which associates to each element x of A, a unique element, denoted by f(x) of B , is called a function or mapping from A to B and we write f : A→B

The element f(x) of B is called the image of x, while x is called the pre-image of f (x).

Let f : A→B, then A is called the domain of f, while B is called the co-domain of f.

The set f(A) = { f (x) : x ∈ A } is called the range of f. - One-one Function: Let f : A→B. If different elements in A have different images in B, then f is said to be a one-one or an injective function or mapping.
- Onto or Surjective Functions: Let f : A→B. If every element in B has at least one preimage in A, then f is said to be an onto function.

If f is onto, then corresponding to each y ∈ B, we must be able to find at least one element x ∈ A such that y = f (x)

Clearly, f is onto if and only if range of f = B - Bijection Function: A one-one and onto function is said to be bijective.

A bijective function is also known as a one-to-one correspondence. - Identity Function: Let A be a non-empty set . Then, the function I defined by

I : A → A : I (x) = x for all x ∈ A is called an identity function on A. - It is a one-to-one onto function with domain A and range A.

Into Functions: Let f : A → B. There exists even a single element in B having no pre-image in A, then f is said to be an into function. - Constant Function: Let f : A → B, defined in such a way that all the elements in A have the same image in B, then f is said to be a constant function.
- Equal Functions: Two functions f and g are said to be equal, written as f = g if they have the same domain and they satisfy the condition f(x) = g(x), for all x.
- Inverse Function: Let f be a one-one onto function from A to B. Let y be an arbitrary element of B. Then f being onto, there exists an element x in A such that f (x) = y.

A function is invertible if and only if f is one-one onto.

- Different types of relations:

Let S = {a, b, c, ….} be any set then the relation R is a subset of the product set S×S

i) If R contains all ordered pairs of the form (a, a) in S×S, then R is called reflexive. In a reflexive relation ‘a’ is related to itself.

For example, ‘Is equal to’ is a reflexive relation for a = a is true.

ii) If (a, b) ∈ R ⇒ (b, a) ∈ R for every a, b∈S then R is called symmetric

For Example a = b⇒b = a. Hence the relation ‘is equal to’ is a symmetric relation.

iii) If (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c)⇒R for every a, b, c, ∈ S then R is called transistive.

For Example a = b, b = c ⇒ a = c. Hence the relation ‘is equal to’ is a transitive relation.

A relation which is reflexive, symmetric and transitive is called an equivalence relation or simply an equivalence. ‘is equal to’ is an equivalence relation.

Similarly, the relation “is parallel to” on the set S of all straight lines in a plane is an equivalence relation. - Domain & Range of a relation: If R is a relation from A to B, then the set of all first coordinates of elements of R is called the domain of R, while the set of all second co-ordinates of elements of R is called the range of R.

So, Dom (R) = {a : (a, b) ∈ R} & Range (R) = {b : (a, b) ∈ R}

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