Question

Consider the following sets, where n > 2:
S1: Set of all n x n matrices with entries from the set {a, 6, c}

S2: Set of all functions from the set {0,1, 2, ..., n2 — 1} to the set {0,1,2}

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

1. There does not exist an injection from S1 to S2.
2. There exists a bijection from S1 to S2
3. There exists a surjection from S1 to S2.
4. There does not exist a bijection from S1 to S2

Answer 1

Correct Answer - Option :

Answer: Option 2 and Option 3 

Concept:

Injection:

It is Mapping/function between two sets A and B (f: A→ B) such that every element in A mapped to a unique element in B. ( one-one function )

Surjection:

It is Mapping/function between two sets A and B (f: A→ B) such that every element in B has a pre-image in A.(onto Function).

Bijection:

It is Mapping/function between two sets A and B such that every element in A is mapped to exactly one element in B and every element in B has an exactly one pre-image in A.

Explanation:

S₁: Set of all n x n matrices with entries from the set {a, b, c}

So in an n× n Matrix total number of the position will be n² and these positions can be filled from any element from the set { a, b, c}

Hence the total number of matrices possible = \({3^{{n^2}}}\)

S₂:Set of all functions from the set {0,1, 2, ..., n2 — 1} to the set {0,1,2}

number of all functions in S₂ will be \({3^{{n^2}}}\).

Option 1:There does not exist an injection from S1 to S2.

This is not correct. Since the cardinality of both the sets S₁ and S₂ is equal. So we can draw one-to-one mapping between Set S₁ and S₂.

Option 2: There exists a bijection from S1 to S₂.

This is correct. Since the cardinality of both the sets S1 and S2 is equal. and hence Bijective function/mapping possible between Set S1 and S2.

Option 3: There exists a surjection from S1 to S2.

This is correct. Since Bijection possible and hence Surjection also possible. Bijection implies Surjection but not vice versa.

Option 4:There does not exist a bijection from S1 to S₂.

This is not correct.