Register

ISI2016-DCG-36

• recategorized by
349 views
0 votes
0 votes

Suppose $X$ and $Y$ are finite sets, each with cardinality $n$.. The number of bijective functions from $X$ to $Y$ is

  1. $n^{n}$
  2. $n\log_{2}n$
  3. $n^{2}$
  4. $n!$
• recategorized by
User Avatar

1 Answer

0 votes
0 votes

Answer $D$

The number of bijective functions are possible $\bf{only\;when}$ the number of elements in both the sets are same.

Its is given that the $X$ and $Y$ are the two sets each with cardinality $n$. $\therefore$ Number of elements are same.

Now, in set $X$

Number of choices for first element = $n$

Number of choices for second element = $n-1$

Number of choices for third element = $n-2$

$\vdots\;\;\;\vdots\;\;\;\vdots$

$\vdots\;\;\;\vdots\;\;\;\vdots$

$\vdots\;\;\;\vdots\;\;\;\vdots$

Number of choices for $(n-1)^{th}$ element = $2$

Number of choices for $n^{th}$ element = $1$

So, total choices are:

$$n(n-1)(n-2)(n-3)\dots\dots3.2.1 = n!$$

$\therefore D$ is the correct option.

 

User Avatar
Voting & Accepting an answer helps improve the community
← PreviousNext →
← Previous in categoryNext in category →

Related questions

0 votes
0 votes
1 answer
1
gatecse asked Sep 18, 2019
312 views
ISI2016-DCG-27
If $A$ be the set of triangles in a plane and $R^{+}$ be the set of all positive real numbers, then the function $f\::\:A\rightarrow R^{+},$ defined by $f(x)=$ area of triangle $x,$ is one-one and into one-one and onto many-one and onto many-one and into
If $A$ be the set of triangles in a plane and $R^{+}$ be the set of all positive real numbers, then the function $f\::\:A\rightarrow R^{+},$ defined by $f(x)=$ area of t...
User Avatar
0 votes
0 votes
0 answers
2
gatecse asked Sep 18, 2019
302 views
ISI2016-DCG-37
Suppose $f_{\alpha}\::\:[0,1]\rightarrow[0,1],\:-1<\alpha<\infty$ is given by $f_{\alpha}(x)=\dfrac{(\alpha+1)x}{\alpha x+1}.$ Then $f_{\alpha}$ is A bijective (one-one and onto) function. A surjective (onto) function. An injective (one-one) function. We can not conclude about the type.
Suppose $f_{\alpha}\::\:[0,1]\rightarrow[0,1],\:-1<\alpha<\infty$ is given by$$f_{\alpha}(x)=\dfrac{(\alpha+1)x}{\alpha x+1}.$$Then $f_{\alpha}$ isA bijective (one-one an...
User Avatar
0 votes
0 votes
1 answer
3
gatecse asked Sep 18, 2019
373 views
ISI2016-DCG-35
Let $A,B$ and $C$ be three non empty sets. Consider the two relations given below: $A-(B-C)=(A-B)\cup C$ $A-(B\cup C)=(A-B)-C$ Both (1) and (2) are correct. (1) is correct but (2) is not. (2) is correct but (1) is not. Both (1) and (2) are incorrect.
Let $A,B$ and $C$ be three non empty sets. Consider the two relations given below:$A-(B-C)=(A-B)\cup C$$A-(B\cup C)=(A-B)-C$Both (1) and (2) are correct.(1) is correct bu...
User Avatar
1 votes
1 votes
1 answer
4
gatecse asked Sep 18, 2019
386 views
ISI2016-DCG-50
The domain of the function $\ln(3x^{2}-4x+5)$ is set of positive real numbers set of real numbers set of negative real numbers set of real numbers larger than $5$
The domain of the function $\ln(3x^{2}-4x+5)$ isset of positive real numbersset of real numbersset of negative real numbersset of real numbers larger than $5$
User Avatar
Link Copied!