Relations and Functions Class 11

Relations and Functions Class 11

Ordered pair

A pair having an order is called ordered pair

Consider (a,b). a is called first component and b is called second component.

If  (a,b) = (c,d), the a = c and b = d

Eg : If (2x-3, 3y+1) = (5,7), find x and y

Ans:

Equating the first component

 2x-3 = 5

2x = 5 + 3 = 8

x = 8/2 = 4

equating the second component

3y+1 = 7

3y = 7-1 = 6

y = 6/3 = 2

Also Read: Trigonometry Equations

Cross product (Cartesian Product)

Q. A × B is the set of all ordered pairs in which first element is from A and second element from B

1.  Let A = {1,2,3} and B = {4,5}, write A×B and B×A?

Ans:

A×B = {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)}

B×A = {(4,1),(4,2),(4,3),(5,1),(5,2),(5,3)}

Note :  If A = ⌽ or B = ⌽-, then A×B = ⌽

n(A×B) = n(A).n(B)

n(A ) = 3, n(B) = 2

n(A×B) = 3×2 = 6

n(A×A) = n(A). n(A) = 3×3= 9

n(A×A×A) = n(A). n(A) .n(A)= 3×3×3= 27

Q. A = {-1,0}. Find A×A×A 

Ans :

A×A = {-1,0}×{-1,0} = {(-1,-1), (-1,0), (0,-1), (0,0)}

A×A×A = {(-1,-1), (-1,0), (0,-1), (0,0)} ×{-1,0}

     = {(-1,-1,-1), (-1,0,-1), (0,-1,-1), (0,0,-1), (-1,-1,1),

   (-1,0,1), (0,-1,1), (0,0,1)}

Q. A = {1,2,3}, B = {3,4}, C = {4,5}, Verify that A×(B⋂C)= (A×B)⋂(A×C) 

Ans :

L H S

B⋂C = {4}

A×(B⋂C) = {1,2,3}×{4}

= {(1,4), (2,4), (3,4)}

R H S

A×B = {1,2,3}×{3,4}

     = {(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)}

A×C = {1,2,3}×{4,5}

     = {(1,4), (1,4),(2,4),(2,5),(3,4),(3,5)}

(A×B)⋂(A×C) = {(1,4),(2,4),(3,4)}

L H S = R H S

      Ans :

3x-y = 0

Put x = 1

3×1 – y = 0

3 – y = 0

Y = 3

Put x = 2, we get y = 6

Put x = 3, we get y = 9

Put x = 4, we get y = 12

Put x = 5, we get y = 15

R = {(1,3),(2,6),(3,9),(4,12)}

Domain = {1,2,3,4}

Range = {3,6,9,12}

Co domain = A

 Q. Let A = {x,y,z}, B = {1,2} find the number of relations from A to B?

Ans:

  m = 3, n= 2

Number of relations = 2mn

= 23×2 = 26 = 64

Q. A = {1,2,3}, B = {4,6,9}. R is a relation from A to B defined by

R = {(x,y) : the difference between x and y is odd}. Write R in roster form. Also write its domain, co domain and range?

Ans :

A×B = {(1,4), (1,6), (1,9), (2,4), (2,6), (2,9), (3,4),(3,6), (3,9)}

R = {(1,4), (1,6), (2,9), (3,4), (3,6)}

Domain = {1,2,3}

Range = {4,6,9}

Co domain = B

Q. A×A has 9 elements of which two elements are (-3,0) and (0,3). R is a Relation on A defined by R = {(x,y) : x + y = 0}. Write r in roster form

Ans :

A×A has 9 elements

A has 3 elements

A = {-3,0,3}

R = {(-3,3), (3,-3), (0,0)}

Q. A = {1,2,3,4}. R is a relation on A defined by R = {(a,b) : b is exactly divisible by a}. Write R in roster form

Ans :

A×A =  {(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)}

  R = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4) }

Q. The figure shows a relation from P to Q.