Name: Jessie James Medel Section:1B BSED SCI
Relation and Function Definition
Relation and function individually are defined as:
Relations - A relation R from a non-empty A to a non-empty set B is a
subset of the cartesian product A × B. The subset is derived by describing a
relationship between the first element and the second element of the ordered
pairs in A × B.
Functions - A relation f from a set A to a set B is said to be a function if
every element of set A has one and only one image in set B. In other words,
no two distinct elements of B have the same pre-image.
Relations and functions can be represented in different forms such as arrow
representation, algebraic form, set - builder, graphical form, roster form, and
tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2)
= 4, f(3) = 9. Now, let us represent this function in different forms.
Set-builder form - {(x, y): y = x2, x ∈ A, y ∈ B}
Roster form - {(1, 1), (2, 4), (3, 9)}
Arrow Representation
Difference Between Relation and Function
Relation Function
A relation in math is a A function is a relation in math
set of ordered pairs such that each element of the
defining the relation domain is related to a single
between two sets. element in the codomain.
Relation Function
A relation may or may
All functions are relations.
not be a function.
Example: {(1, x), (1, y),
Example: {(1, x), (6, y), (4, z)}
(4, z)}
Types of Relations
Given below is a list of different types of relations:
Empty Relation - A relation is an empty relation if it has no elements, that is,
no element of set A is mapped or linked to any element of A. It is denoted by
R = ∅.
Universal Relation - A relation R in a set A is a universal relation if each
element of A is related to every element of A, i.e., R = A × A. It is called the
full relation.
Identity Relation - A relation R on A is said to be an identity relation if each
element of A is related to itself, that is, R = {(a, a) : for all a ∈ A}
Inverse Relation - Define R to be a relation from set P to set Q i.e., R ∈ P × Q.
The relation R-1 is said to be an Inverse relation if R-1 from set Q to P is
denoted by R-1 = {(q, p): (p, q) ∈ R}.
Reflexive Relation - A binary relation R defined on a set A is said to be
reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R.
Symmetric Relation - A binary relation R defined on a set A is said to be
symmetric if and only if, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈
R, then we must have bRa, that is, (b, a) ∈ R.
Transitive Relation - A relation R is transitive if and only if (a, b) ∈ R and (b,
c) ∈ R ⇒ (a, c) ∈ R for a, b, c ∈ A
Equivalence Relation - A relation R defined on a set A is said to be an
equivalence relation if and only if it is reflexive, symmetric and transitive.
Antisymmetric Relation - A relation R on a set A is said to be antisymmetric
if (a, b) ∈ R and (b , a) ∈ R ⇒ a = b.
Types of Functions
Given below is a list of different types of functions:
One-to-One Function - A function f: A → B is said to be one-to-one if each
element of A is mapped to a distinct element of B. It is also known
as Injective Function.
Onto Function - A function f: A → B is said to be onto, if every element of B
is the image of some element of A under f, i.e, for every b ∈ B, there exists an
element a in A such that f(a) = b. A function is onto if and only if the range of
the function = B.
Many to One Function - A many to one function is defined by the function f:
A → B, such that more than one element of the set A are connected to the
same element in the set B.
Bijective Function - A function that is both one-to-one and onto function is
called a bijective function.
Constant Function - The constant function is of the form f(x) = K, where K is
a real number. For the different values of the domain(x value), the same
range value of K is obtained for a constant function.
Identity Function - An identity function is a function where each element in
a set B gives the image of itself as the same element i.e., g (b) = b ∀ b ∈ B.
Thus, it is of the form g(x) = x.
Algebraic functions are based on the degree of the algebraic expression. The
important algebraic functions.
Relations and Functions Examples
Example 1: Given three relations R, S, T from A = {x, y, z} to B = {u, v, w} defined
as: 1) R = {(x, u), (z, v)}, 2) S = {(x, u), (y, v), (z, w)}, 3) T = {(x, u), (x, v), (z, w)}. Identify
which of the given relations is/are function(s) using relations and functions
definition.
Solution: Let us check each part one by one.
1) For R = {(x, u), (z, v)}, each element of A is not mapped to an element of B which
violates the definition of a function. Hence, R is not a function.
2) For S = {(x, u), (y, v), (z, w)}, each element of A is mapped to a unique element of
B which satisfies the definition of a function. Hence, S is a function.
WHAT IS RANGE?
The range is the difference between the smallest and highest numbers in a list or
set. To find the range, first put all the numbers in order. Then subtract (take away)
the lowest number from the highest. The answer gives you the range of the list.
Example 1: In {4, 6, 9, 3, 7} the lowest value is 3, and the highest is 9. So the
range is 9 − 3 = 6.
Example 2: if the given data set is {2,5,8,10,3}, then the range will be 10 – 2 = 8.
WHAT IS DOMAIN?
The domain of a function is the set of values that we are allowed to plug into our
function. This set is the x values in a function such as f(x). The range of a function
is the set of values that the function assumes. This set is the values that the
function shoots out after we plug an x value in.
Commonly considered types of domains are domains
with continuous boundary, lipschitz bounderies, C1 boundary, and so forth.
A bounded domain or bounded region is that which is a bounded set, i.e., having
a finite measure. An exterior domain or external domain is the interior of
the complement of a bounded domain.
WHAT IS SET OPERATION?
Set operations can be defined as the operations that are performed on two or more
sets to obtain a single set containing a combination of elements from both all the
sets being operated upon. There are basically three types of operation on sets in
Mathematics; they are: The Union of Sets (∪) The Intersection of Sets (∩)
The Union of Sets (∪)In Operation set in math, the union of sets can be described
as the set that contains all the elements of all the sets on which the union
operation is applied. The union of sets can be denoted by the symbol ∪. The set
formed by the union of P and Q will contain all the elements of set P and set Q
together. The union of sets can be interpreted as:P ∪ Q = n(P) + n(Q)Where n(P)
represents, the cardinal number of set P andn(Q) represents the cardinal number of
set QLet’s take an example:Set P- {1, 2, 3, 4, 5} and Set Q- {7, 8, 9, 10}Therefore, P
∪ Q = {1, 2, 3, 4, 5, 7, 8, 9, 10}
The Intersection of Sets (∩)The intersection of sets is referred to as a set
containing the elements that are common to all the sets being operated upon. It is
denoted by the symbol ∩. The set that is formed by the intersection of both the sets
will contain all the elements that are common in set P as well as Set Q.Let’s take an
example:Set P= {1, 2, 3, 4} and Set Q= {3, 4, 5, 6}Then, P ∩ Q = {3, 4}
Properties of Set Operations
There are certain properties of set operations; these properties are used for set
operations proofs. The properties are as follows:
Distributive Property
P ∩ (Q ∪ R) = (P ⋂ Q) ∪ (P ∩ R)P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R)
Commutative Property
P ∪ Q = Q ∪ PP ∩ Q = Q ∩ P
WHAT IS LOGIC?
Logic means reasoning. The reasoning may be a legal opinion or mathematical
confirmation. We apply certain logic in Mathematics. Basic Mathematical logics are
a negation, conjunction, and disjunction
SIMPLE STATEMENT
A simple statement is a statement which has one subject and one predicate.
What is problem solving and reasoning in mathematics?
Reasoning in maths is the process of applying logical thinking to a situation to
derive the correct problem solving strategy for a given question, and using this
method to develop and describe a solution. Put more simply, mathematical
reasoning is the bridge between fluency and problem solving.
What is inductive reasoning?
Inductive reasoning is the process of reaching a general conclusion by examining
specific examples. The conclusion formed by using inductive reasoning is often
called a conjecture, since it may or may not be correct.
Example. Use inductive reasoning to predict the most probable next number in
each of the following lists. (a) 3, 6, 9, 12, 15, ? (b) 1, 3, 6, 10, 15,
What is deductive reasoning?
Deductive reasoning is a type of deduction used in science and in life. It is when
you take two true statements, or premises, to form a conclusion. For example, A is
equal to B. B is also equal to C. Given those two statements, you can conclude A is
equal to C using deductive reasoning.
Everyday life often tests our powers of deductive reasoning. Did you ever wonder
when you'd need what you learned in algebra class?
Well, if nothing else, those lessons were meant to stretch our powers of deductive
reasoning. Remember, if a = b and b = c, then a = c. Let's flesh that out with added
examples:
All numbers ending in 0 or 5 are divisible by 5. The number 35 ends with a
5, so it must be divisible by 5.
All birds have feathers. All robins are birds. Therefore, robins have feathers.
What is Recreational problem?
Recreational mathematics is mathematics carried out for recreation (entertainment)
rather than as a strictly research- and application-based professional activity or as
a part of a student's formal education.
Examples of recreational math could include classic games such as Monopoly or
any number of card games requiring addition and subtraction.
Name: CHRISTIAN V. DELOS TRAYCO
Section: BSED SCIENCE 1B
Relation and Function Definition
Relation and function individually are defined as:
Relations - A relation R from a non-empty A to a non-empty set B is a
subset of the cartesian product A × B. The subset is derived by describing a
relationship between the first element and the second element of the ordered
pairs in A × B.
Functions - A relation f from a set A to a set B is said to be a function if
every element of set A has one and only one image in set B. In other words,
no two distinct elements of B have the same pre-image.
Relations and functions can be represented in different forms such as arrow
representation, algebraic form, set - builder, graphical form, roster form, and
tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2)
= 4, f(3) = 9. Now, let us represent this function in different forms.
Set-builder form - {(x, y): y = x2, x ∈ A, y ∈ B}
Roster form - {(1, 1), (2, 4), (3, 9)}
Arrow Representation
Difference Between Relation and Function
Relation Function
A relation in math is a A function is a relation in math
set of ordered pairs such that each element of the
defining the relation domain is related to a single
between two sets. element in the codomain.
A relation may or may
All functions are relations.
not be a function.
Example: {(1, x), (1, y),
Example: {(1, x), (6, y), (4, z)}
(4, z)}
Types of Relations
Given below is a list of different types of relations:
Empty Relation - A relation is an empty relation if it has no elements, that is,
no element of set A is mapped or linked to any element of A. It is denoted by
R = ∅.
Universal Relation - A relation R in a set A is a universal relation if each
element of A is related to every element of A, i.e., R = A × A. It is called the
full relation.
Identity Relation - A relation R on A is said to be an identity relation if each
element of A is related to itself, that is, R = {(a, a) : for all a ∈ A}
Inverse Relation - Define R to be a relation from set P to set Q i.e., R ∈ P × Q.
The relation R-1 is said to be an Inverse relation if R-1 from set Q to P is
denoted by R-1 = {(q, p): (p, q) ∈ R}.
Reflexive Relation - A binary relation R defined on a set A is said to be
reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R.
Symmetric Relation - A binary relation R defined on a set A is said to be
symmetric if and only if, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈
R, then we must have bRa, that is, (b, a) ∈ R.
Transitive Relation - A relation R is transitive if and only if (a, b) ∈ R and (b,
c) ∈ R ⇒ (a, c) ∈ R for a, b, c ∈ A
Equivalence Relation - A relation R defined on a set A is said to be an
equivalence relation if and only if it is reflexive, symmetric and transitive.
Antisymmetric Relation - A relation R on a set A is said to be antisymmetric
if (a, b) ∈ R and (b , a) ∈ R ⇒ a = b.
Types of Functions
Given below is a list of different types of functions:
One-to-One Function - A function f: A → B is said to be one-to-one if each
element of A is mapped to a distinct element of B. It is also known
as Injective Function.
Onto Function - A function f: A → B is said to be onto, if every element of B
is the image of some element of A under f, i.e, for every b ∈ B, there exists an
element a in A such that f(a) = b. A function is onto if and only if the range of
the function = B.
Many to One Function - A many to one function is defined by the function f:
A → B, such that more than one element of the set A are connected to the
same element in the set B.
Bijective Function - A function that is both one-to-one and onto function is
called a bijective function.
Constant Function - The constant function is of the form f(x) = K, where K is
a real number. For the different values of the domain(x value), the same
range value of K is obtained for a constant function.
Identity Function - An identity function is a function where each element in
a set B gives the image of itself as the same element i.e., g (b) = b ∀ b ∈ B.
Thus, it is of the form g(x) = x.
Algebraic functions are based on the degree of the algebraic expression. The
important algebraic functions.
Relations and Functions Examples
Example 1: Given three relations R, S, T from A = {x, y, z} to B = {u, v, w} defined
as: 1) R = {(x, u), (z, v)}, 2) S = {(x, u), (y, v), (z, w)}, 3) T = {(x, u), (x, v), (z, w)}. Identify
which of the given relations is/are function(s) using relations and functions
definition.
Solution: Let us check each part one by one.
1) For R = {(x, u), (z, v)}, each element of A is not mapped to an element of B which
violates the definition of a function. Hence, R is not a function.
2) For S = {(x, u), (y, v), (z, w)}, each element of A is mapped to a unique element of
B which satisfies the definition of a function. Hence, S is a function.
WHAT IS RANGE?
The range is the difference between the smallest and highest numbers in a list or
set. To find the range, first put all the numbers in order. Then subtract (take away)
the lowest number from the highest. The answer gives you the range of the list.
Example 1: In {4, 6, 9, 3, 7} the lowest value is 3, and the highest is 9. So the
range is 9 − 3 = 6.
Example 2: if the given data set is {2,5,8,10,3}, then the range will be 10 – 2 = 8.
WHAT IS DOMAIN?
The domain of a function is the set of values that we are allowed to plug into our
function. This set is the x values in a function such as f(x). The range of a function
is the set of values that the function assumes. This set is the values that the
function shoots out after we plug an x value in.
Commonly considered types of domains are domains
with continuous boundary, lipschitz bounderies, C1 boundary, and so forth.
A bounded domain or bounded region is that which is a bounded set, i.e., having
a finite measure. An exterior domain or external domain is the interior of
the complement of a bounded domain.
WHAT IS SET OPERATION?
Set operations can be defined as the operations that are performed on two or more
sets to obtain a single set containing a combination of elements from both all the
sets being operated upon. There are basically three types of operation on sets in
Mathematics; they are: The Union of Sets (∪) The Intersection of Sets (∩)
The Union of Sets (∪)In Operation set in math, the union of sets can be described
as the set that contains all the elements of all the sets on which the union
operation is applied. The union of sets can be denoted by the symbol ∪. The set
formed by the union of P and Q will contain all the elements of set P and set Q
together. The union of sets can be interpreted as:P ∪ Q = n(P) + n(Q)Where n(P)
represents, the cardinal number of set P andn(Q) represents the cardinal number of
set QLet’s take an example:Set P- {1, 2, 3, 4, 5} and Set Q- {7, 8, 9, 10}Therefore, P
∪ Q = {1, 2, 3, 4, 5, 7, 8, 9, 10}
The Intersection of Sets (∩)The intersection of sets is referred to as a set
containing the elements that are common to all the sets being operated upon. It is
denoted by the symbol ∩. The set that is formed by the intersection of both the sets
will contain all the elements that are common in set P as well as Set Q.Let’s take an
example:Set P= {1, 2, 3, 4} and Set Q= {3, 4, 5, 6}Then, P ∩ Q = {3, 4}
Properties of Set Operations
There are certain properties of set operations; these properties are used for set
operations proofs. The properties are as follows:
Distributive Property
P ∩ (Q ∪ R) = (P ⋂ Q) ∪ (P ∩ R)P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R)
Commutative Property
P ∪ Q = Q ∪ PP ∩ Q = Q ∩ P
WHAT IS LOGIC?
Logic means reasoning. The reasoning may be a legal opinion or mathematical
confirmation. We apply certain logic in Mathematics. Basic Mathematical logics are
a negation, conjunction, and disjunction
SIMPLE STATEMENT
A simple statement is a statement which has one subject and one predicate.
What is problem solving and reasoning in mathematics?
Reasoning in maths is the process of applying logical thinking to a situation to
derive the correct problem solving strategy for a given question, and using this
method to develop and describe a solution. Put more simply, mathematical
reasoning is the bridge between fluency and problem solving.
What is inductive reasoning?
Inductive reasoning is the process of reaching a general conclusion by examining
specific examples. The conclusion formed by using inductive reasoning is often
called a conjecture, since it may or may not be correct.
Example. Use inductive reasoning to predict the most probable next number in
each of the following lists. (a) 3, 6, 9, 12, 15, ? (b) 1, 3, 6, 10, 15,
What is deductive reasoning?
Deductive reasoning is a type of deduction used in science and in life. It is when
you take two true statements, or premises, to form a conclusion. For example, A is
equal to B. B is also equal to C. Given those two statements, you can conclude A is
equal to C using deductive reasoning.
Everyday life often tests our powers of deductive reasoning. Did you ever wonder
when you'd need what you learned in algebra class?
Well, if nothing else, those lessons were meant to stretch our powers of deductive
reasoning. Remember, if a = b and b = c, then a = c. Let's flesh that out with added
examples:
All numbers ending in 0 or 5 are divisible by 5. The number 35 ends with a
5, so it must be divisible by 5.
All birds have feathers. All robins are birds. Therefore, robins have feathers.
What is Recreational problem?
Recreational mathematics is mathematics carried out for recreation (entertainment)
rather than as a strictly research- and application-based professional activity or as
a part of a student's formal education.
Examples of recreational math could include classic games such as Monopoly or
any number of card games requiring addition and subtraction.
