If A has 5 elements and B has 4 elements then number of relations from A to B are

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Sum

If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A × B)?

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Solution

It is given that set A has 3 elements and the elements of set B are 3, 4, and 5.

⇒ Number of elements in set B = 3

Number of elements in (A × B)

= (Number of elements in A) × (Number of elements in B)

= 3 × 3 = 9

Thus, the number of elements in (A × B) is 9.

Concept: Cartesian Product of Sets

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Chapter 2: Relations and Functions - Exercise 2.1 [Page 33]

Q 2Q 1Q 3

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NCERT Class 11 Mathematics

Chapter 2 Relations and Functions
Exercise 2.1 | Q 2 | Page 33

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If set A has p no. of elements and set B has q number of elements then the total number of relations defined from set A to set B is 2pq.

Math Formulas

If set A has p no. of elements and set B has q number of elements then the total numberof relations defined from set A to set B is 2pq.

Definition :-For any set A such that n(A) = n

Then number of all relations on A is

As the total number of relations that can be defined a set A to B is the number of possible subsets of A ×B. If n(A) = p and n(B) = q then n(A × B) = pq and the number of subsets of

Example 1 :-If set

Solution :-
Set A has 2 elements and set B has 3 elements then the no. of relation defined from A to B is

Example 2 :-The number of relations defined on set A = {a,b,c,d}.

Solution :-
Set A has 4 elements
No. of relation defined on set A is

Answer

Verified

Hint: A General Function points from each member of "A" to a member of "B". It never has one "A" pointing to more than one "B", so one-to-many is not OK in a function (so something like "f(x) = 7 or 9" is not allowed)But more than one "A" can point to the same "B" (many-to-one is OK).Injective means we won't have two or more "A"s pointing to the same "B". So many-to-one is NOT OK (which is OK for a general function). As it is also a function one-to-many is not OK But we can have a "B" without a matching "A" Injective is also called "One-to-One".

Complete step by step solution: The Set A has 4 elements and the Set B has 5 elements and we have to find the number of injective mappings.
Let f be such a function.
 Now, f takes inputs from set A whereas the output value of f comes from set B.
Using the fact that injective functions are one-one and onto,
\[f(1)\;\]can take \[5\] values,
\[\;f(2)\;\]can then take only \[4\;\]values ,
\[\;f(3)\;\]can take only \[3\;\]and
\[\;f(4)\;\]only \[2\].
Hence the total number of functions are \[5 \times 4 \times 3 \times 2 = 120\].

Note: In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its co-domain. In simple words, every element of the function’s co-domain is the image of at most one element of its domain.

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