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MATHS TRICKS FOR RELATION AND FUNCTION

maths trips and trick for iit jee relation and function

MATHS TRICKS FOR RELATION AND FUNCTION

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Hey Students, in this post, you will learn about the Maths Tips and Tricks for IIT JEE Relation and Function chapter from class 12 Maths Subjects. This maths tips and tricks is very useful in solving the problems of IIT JEE Mains and Advance entrance exams. 

Since we already know that the chapter of relation and function contains conceptual questions that required more time to solve. This maths tips and tricks reduces the time consuming problems. By using this maths tips and tricks, you will able to solve more questions in the IIT JEE examination hall in very short time. So read this post carefully. 

Maths Tips and Tricks 1: If A and B are any two non-empty sets having n elements in common, then A\times B and B\times Ahave n ^{2} elements in common.

Example : A and B are two sets having 4 elements in common. If n(A) = 5 and n(B) = 9, then n[(A\times B\cap (B\times A)] =

(A) 9                           (B) 3                          (C) 16                                         (D)  20

Solution:  The number of common elements in A and B is 3. Thus, n[(A\times B\cap (B\times A)] = (3) ^{2}=9.

Therefore, Ans:(A)

Maths Tips and Tricks 2: Let A and B be two non-empty finite sets consisting of m and n elements respectively, then total number relation from A to B is 2 ^{mn}.

Example: If A = {2,3,5,6,8} and B = {1,5,9,8,7,4,3}, then find the total number of relation from A to B is

(A) 2 ^{10}                           (B) 2 ^{25}                        (C) 2 ^{35}                                         (D) 2 ^{20}

Solution: Since n(A) = m = 5 and n(B) = n = 7. Thus, the total number of the relation from A to B is 2 ^{5\times 7}= 2^{35}}.

Therefore, Ans (C)

Maths Tips and Tricks 3: If  a set A has n elements, then the number of reflexive relations from A to A

is 2 ^{n(n-1)}.

Example: If  A = {x:x is a natural number and x <= 20}, then the number of reflexive relation from set A to itself is

(A) 2 ^{180}                        (B) 2 ^{290}                       (C)  2 ^{380}                                          (D) none

Solution: Since A = {x:x is a natural number and x <= 20} \Rightarrow A = {1,2,3,…,18,19,20}\Rightarrow n(A) = 20

Thus, the number of the reflexive relation = 2 ^{20(20-1))}=2 ^{380}

Therefore, Ans (C)

Maths Tips and Tricks 4: Let A be a finite set containing n elements. Then, the total number of binary operation on A is n ^{n ^{2}}

Example: If  A = {x : x is first six even multiple of 4}, then the number of binary operation on A is

(A) 6 ^{36}                        (B) 6 ^{30}                       (C) 6 ^{20}                                          (D) none

Solution: Since A = {x:x is first six even multiple of 4} \Rightarrow A = {4,8,12,16,20,24}\Rightarrow n(A) = 6

Thus, the number of the binary operation on A  = 6 ^{6^{2}}=6 ^{36}

Therefore, Ans (A)

Maths Tips and Tricks 5: Let A be a finite set containing n elements. Then, the total number of commutative binary operation on A is [\frac{n(n+1)}{{2}}]

Example: If  A = {x : x is a natural number such that 4x-7\leq 21}, then the number of commutative binary operation on A is

(A) 18                     (B) 20                       (C) 30                                          (D) 28

Solution: Since A = {x : x is a natural number such that 4x-7\leq 21\Rightarrow A = {1,2,3,4,5,6,7}\Rightarrow n(A) = 7

Thus, the number of the commutative binary operation on A  = \frac{7(7+1)}{2}=28

Therefore, Ans (D)

Maths Tips and Tricks 6: If a set contains m elements and another set B contains n elements, then the total number of functions from the set A to B is (n) ^{m}.

Example: If  A = {1,5,9,7,14,22} and B = {2,3,5,6}, then the number of function from the set A to B is

(A) 2048                     (B) 4096                       (C) 1522                                          (D)1024

Solution: Since A = {1,5,9,7,14,22}\Rightarrow n(A) = 6 and B = {2,3,5,6} \Rightarrow n(B) = 4

Thus, the total number of the function A to B is  (4) ^{6}=4096

Therefore, Ans (B)

Maths Tips and Tricks 7: Let a function f defined as A\rightarrow B such that A and B are finite sets having m and n elements respectively. The number of one-one function is _{}^{n}\textrm{P_{{m}}}, for n\geq m and O (zero) for n < m.

Example: If  A = {6,7,8,9,10,11,12} and B = {2,3,4,5,6} and f:A\rightarrow B, then the total number of one-one function is

(A) 3024                    (B) 2863                      (C) 2520                                         (D)3042

Solution: Since A = {6,7,8,9,10,11,12}\Rightarrow n(A) = 7 and B = {2,3,4,5,6} \Rightarrow n(B) = 5 and n(A) > n(B).

Thus, the total number of one-one function from A to B is  _{}^{7}\textrm{P_{5}}=2520

Therefore, Ans (C)

Maths Tips and Tricks 8: Let a function f defined as A\rightarrow B such that A and B are finite sets having m and n elements respectively. The number of many-one function is  n^{m}-_{}^{n}\textrm{P_{m}}  _{}^{n}\textrm{P_{{m}}}, for n\geq m and (n) ^{m} for n < m.

Example: If  A = {1,3,5,7,9} and B = {a,e,i,o,u} and f:A\rightarrow B, then the total number of many-one function is

(A) 3005                    (B) 3105                      (C) 3250                                         (D)3025

Solution: Since A = {1,3,5,7,9}\Rightarrow n(A) = 5 and B = {a,e,i,o,u} \Rightarrow n(B) = 5 and n(A) = n(B).

Thus, the total number of many-one function A\rightarrow B is  5^{5}-_{}^{5}\textrm{P_{5}} =3125-120=3005

Therefore, Ans (A)

Maths Tips and Tricks 9: Let a function f defined as A\rightarrow B such that A and B are finite sets having m and n elements respectively. The number of onto function is

(i)  n^{m}-_{}^{n}\textrm{C_{1}} (n-1)^m+_{}^{n}\textrm{C_{2}} (n-2)^m-_{}^{n}\textrm{C_{3}} (n-3)^m+..., for  n<m

(ii) n! for n=m

(iii) 0 for n>m

Example: If  A = {2,4,6,8} and B = {a,b,c,d} and f:A\rightarrow B, then the total number of onto function is

(A) 16                    (B) 18                     (C) 22                                         (D)24

Solution: Since A = {2,4,6,8}\Rightarrow n(A) = 4 and B = {a,b,c,d} \Rightarrow n(B) = 4 and n(A) = n(B).

Thus, the total number of onto function A\rightarrow B is 4!=24

Therefore, Ans (D)

Maths Tips and Tricks 10: Let a function f defined as A\rightarrow B such that A and B are finite sets having m and n elements respectively. The number of into function is

(i)  _{}^{n}\textrm{C_{1}} (n-1)^m-_{}^{n}\textrm{C_{2}} (n-2)^m+_{}^{n}\textrm{C_{3}} (n-3)^m-..., for  n\leq m

(ii) (n) ^{m} for n > m

Example: If  A = {1,2,3} and B = {a,b,c,d,e,f,g,h} and f:A\rightarrow B, then the total number of into function is

(A) 243                    (B) 512                    (C)6561                                         (D)1024

Solution: Since A = {1,2,3} \Rightarrow n(A) = 3 and B = {a,b,c,d,e,f,g,h} \Rightarrow n(B) = 8 and n(A) < n(B).

Thus, the total number of into function A\rightarrow B is (8) ^{3}=512

Therefore, Ans (C)

Maths Tips and Tricks 11: Let a function f defined as A\rightarrow B such that A and B are finite sets having m and n elements respectively. The number of constant function is n.

Example: If  A = {1,2,3} and B = {a,b,c,d,e,f,g,h} and f:A\rightarrow B, then the total number of constant function is

(A) 3                    (B) 8                    (C)10                                         (D)16

Solution: Since A = {1,2,3} \Rightarrow n(A) = 3 and B = {a,b,c,d,e,f,g,h} \Rightarrow n(B) = 8

Thus, the total number of constant function A\rightarrow B is 8.

Therefore, Ans (B)

Maths Tips and Tricks 12: If A and B are finite sets and f:A\rightarrow B is a bijection (One One Onto), then A and B have same number of the elements. If A has n elements, then the number of the bijections from A to B is n!

Example: If  A = {1,2,3,4,5} , then the total number of bijection function is f:A\rightarrow A is

(A) 110                     (B) 115                    (C)120                                         (D)125

Solution: Since A = {1,2,3,4,5} \Rightarrow n(A) = 5

Thus, the total number of bijection function is f:A\rightarrow A is 5! = 120.

Therefore, Ans (C)

Maths Tips and Tricks 13: If f(x) is periodic with period T, then kf(ax+b) is periodic with period \frac{T}{|a|}, where a, b, k \in R and a, k \neq 0.

Example: The period of tan8x is

(A) \Pi                    (B) 2\Pi                   (C)\frac{\Pi }{8}                                     (D)\frac{\Pi }{12}

Solution: Since tanx is a periodic function with period \Pi , then the period of tan8x is \frac{\Pi }{8}

Therefore, Ans (C)

So, use these Maths Tricks based on Relations and Functions chapter in competitive engineering exams, which helps you to solve tedious calculation based problems in IIT JEE MAINS and ADVANCE and other Engineering Entrance Exams.

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