In Algebra 1,Functions and relations are the part of algebra which includes functions problems and simplifying functions and relational expressions.
Cartesian Product of Two Sets :
In Cartesian coordinate system ,Suppose A and B be two non-empty sets. The set containing all the ordered pairswhere the first element is taken from A and second element is taken from B is called the Cartesian product of two sets A and B . It is denoted by A X B and read as A cross B .
A X B = { (x , y)/ x $\in$ A and y $\in$ B }.
Example :
Q1 : If A= { 1,2,3 } and B = { p, q } find the Cartesian product A X B .
Solution : A X B = { (1, p), (1, q), (2, p), (2, q), (3, p), (3, q) }
Note : If n(A)=p , n(B)=q elements then number of elements in n(A X B) = pq .
Relations :
A Relation is a subset of a Cartesian Product of two sets . Therefore relation is set of ordered pairs.
Let A and B be two sets . Let R be a relation from A into B then R$\subseteq$A X B.
A is called domain of R and B is called Co-domain of R.
Note : 1) If A has p elements and B has q elements then the types of relations from A into B is 2pq .
2) (a, b) = (b , a) unless a = b .
3) If (a, b) = (p, q) then a = p and b = q .
Example:
Q2: Let R be a relation from A= { 1, 2, 3 } into B = { 2, 3 } defined by " x less than y " where x $\in$ A and y$\in$ B, find R ?
Solution : R = {(1, 2), (1, 3), (2, 3) }.
Q3 : If (a, b) = ( 5, 8) find a and b values.
Solution: a = 5 and b = 8.
Q4 : If ( a, 2) =(3, b) then find a and b ?
Solution : a= 3 and b=2.
Q5: If ( x+y , 1) = (3, y-x ) then find x and y ?
Solution : x + y = 3 .......(1) and
1 = y - x that is y - x = 1 ..........(2)
adding (1) and (2) equations we get,
x + y + y - x = 3 + 1
2 y = 4 then y = 2
substituting y= 2 in (1) we get
x + 2 = 3 then x = 1
Therefore x = 1 and y = 2.
Q6 : If ( a+b , a-b) = (5, 1) then find the values of a and b ?
Solution : a + b = 5 ...........(1)
a - b = 1 ............(2)
adding (1) and (2) we get ,
2 a = 6 then a = 3.
substituting a = 3 in (1) we get,
3 + b = 5 then b = 5 - 3 = 2.
Therefore a = 3 and b = 2 .
Functions and Relations Types
Relations are of 4 types. They are i) One-to-One relation ii) Many-to-One relation iii) One-to-Many relation iv) Many-to-Many relation.
i) One-to-One relation :
A relation R : A ? B is said to be One-to-One relation if no two elements of A have the same image in B.
Example:
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ii) Many-to-One Relation :
A relation R : A ? B is said to be many-to-One relation if two or more elements of A are related to an element of B .
Example :
iii) One-to-Many Relation :
A relation R : A ? B is said to be One-to-Many relation if an element of A is related to two or more elements of B.
Example :
iv) Many-to-Many Relation :
A relation R : A ? B is said to be Many-to-Many relation if two or more elements of A are related to two or more elements of B.
Example :
What is a Relation
We can describe a relation in five ways . They are
1. List form or Roster form 2. Set builder form 3. Arrow diagrams 4. Tree diagrams 5. Graphical representation.
1. List form or Roster form :
In this method we list all the ordered pairs that satisfy the formula given in relation .
Example :
Q: If A = { 1, 2, 3 } write E is the relation having the property ' is less than ' for its elements.
Solution : Given A = { 1, 2, 3 }
Then A X A = { (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3) }
Given E is the relation consisting all those ordered pairs whose first coordinates are less than second coordinates.
Therefore E = { (1, 2), (1, 3), (2, 3) }.
2. Set builder form :
In this method we describe the relation by statig the property that connects the first and second coordinates of every ordered pairs of the relation .
Example :
Q1 : If A = {1, 2, 3 } and L = { (1, 1), (2, 2), (3, 3) } . Write L in set builder form.
Solution : L = { (1, 1), (2, 2), (3, 3) }
L = { (x, y)/(x, y) $\in$ A X A , x = y }
Q2 : Suppose A , B are two sets , describe the elements of A X B and B X A in i) Arrow diagram ii)Tree diagram iii) Graphical representation, where A = { 1, 4, 5 } B = {2 , 4 } . What do you notice?
Solution : A = { 1, 4, 5 } , B ={ 2, 4 }
A X B = { 1, 4, 5 } X { 2, 4 }
A X B = { ( 1, 2), (1, 4), (4, 2), (4, 4), (5, 2), (5, 4 ) }
B X A = { 2, 4 } X { 1, 4, 5 }
B X A = { (2, 1), (2, 4), (2, 5), (4, 1), (4, 4), (4, 5) }
We notice that A X B ? B X A
i) Arrow diagram of A X B
Arrow diagram of B X A
ii) Tree diagram of A X B
Tree diagram of B X A.
iii) Graphical representation of A X B.
Graphical representation of B X A.
Sets Examples
This following algebra 1 problems gives a step by step procedure to solve the given questions
Q1 : If A = { 1, 2, 3 }, B= { 3, 4, 5 } and C = { 4, 6 } . Find the following .
i) A X (B ? C) ii) (A X B) ? (A X C ) iii) A X ( B ? C ) iv) (A X B) ? ( A X C) .
What inferences can you make from these resultant sets ?
solution : Given A = {1, 2, 3 }, B = {3, 4, 5 } and C = {4, 6 }
B ? C = {3, 4, 5 } ? {4, 6 }
= {3, 4, 5, 6 }
B ? C = {3, 4, 5 } ? {4. 6 }
= {4}
A X B = {1, 2, 3 } X {3, 4, 5 }
= { (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5) }
A X C = {1, 2, 3 } X {4, 6 }
= { (1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6) }
A X (B ? C) = {1, 2, 3 } X {3, 4, 5, 6 }
= {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6) }
(A X B) ? (A X C) = { (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5) } U { (1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6) }
(A X B) ? (A X C) ={(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6), }
we observe that A X (B ? C) = (A X B) ? (A X C)
similarly A X ( B ? C ) = { 1, 2, 3 } X {4}
= { (1, 4), (2, 4), (3, 4) }
(A X B) ? ( A X C) = { (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5) } ? { (1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6) }
(A X B) ? ( A X C) = { (1, 4), (2, 4), (3, 4) }
Therefore A X ( B ? C ) = (A X B) ? ( A X C)
Related Tags
Relations and functions , Functions and it's relations , function relations