GATE CSE 2021 Set 2 | Question: 11

Question

Consider the following sets, where $n \geq 2$:

• $S_1$: Set of all $n \times n$ matrices with entries from the set $\{ a, b, c\}$
• $S_2$: Set of all functions from the set $\{0,1,2, \dots, n^2-1\}$ to the set $\{0, 1, 2\}$

Which of the following choice(s) is/are correct?

• A. There does not exist a bijection from $S_1$ to $S_2$
• B. There exists a surjection from $S_1$ to $S_2$
• C. There exists a bijection from $S_1$ to $S_2$
• D. There does not exist an injection from $S_1$ to $S_2$

Comments

Can we show this in diagrammatic form?

Answer 1

$S_1:$ There are $n^2$ element in the matrix, we have $3$ choices for each element, so number of such matrices $=3^{n^2}.$

$S_2:$ There are $n^2$ total elements with $3$ choices for each element, so number of functions possible $=3^{n^2}.$

As the cardinality of both the sets are same, we can establish a bijection from one set to another. As bijection is possible, surjection is also possible.

So options B and C.

Answer 2

see

lets take n=3

then accroding to statment 1 we have 3*3 matrix  in which we have 9 entries  no of such matrix 3^9

and statment2 {0,1,2,3,4,5,6,7,8}  fxn {0,1,2}  no of fxn possible 3^9

so cardinilaties same there may exist bijection and surjection  both

B,C

Answer 3

A bijection between two sets is possible iff the cardinality of both sets are equal. 

So,here since both sets have same cardinality,bijection is possible, and hence surjection and injection too

Option B and C r right