Virtual Labs

Permutations and Permutation Groups

Which of these is an injective but not a bijective function?

f:A → A

where

A=\{1,2,3,4,5\}

and

f(a)=a

for all

a∈ A

Explanation

f:A → B

where

A=\{1,2,3\}

B=\{1,2,3,4,5,6\}

and

f(a)=2a

for

a∈ A

Explanation

f:A → B

where

A=\{1,2,3,4,5\}

B={1}

and

f(a)=1

for all

a∈ A

Explanation

f:A → A

where

A=\{1,2,3,4,5\}

and

f(a)=1

for all

f(a),a∈ A

Explanation

Which of these is an surjective but not a bijective function?

f:A → A

where

A=\{1,2,3,4,5\}

and

f(a)=a

for all

a∈ A

Explanation

f:A → B

where

A=\{1,2,3\}

B=\{1,2,3,4,5,6\}

and

f(a)=2a

for

a∈ A

Explanation

f:A → B

where

A=\{1,2,3,4,5\}

B=\{1\}

and

f(a)=1

for all

a∈ A

Explanation

f:A → A

where

A=\{1,2,3,4,5\}

and

f(a)=1

for all

f(a),a∈ A

Explanation

Which of these is a bijective function?

f:A → A

where

A=\{1,2,3,4,5\}

and

f(a)=a

for all

a∈ A

Explanation

f:A → B

where

A=\{1,2,3\}

B=\{1,2,3,4,5,6\}

and

f(a)=2a

for

a∈ A

Explanation

f:A → B

where

A=\{1,2,3,4,5\}

B=\{1\}

and

f(a)=1

for all

a∈ A

Explanation

f:A → A

where

A=\{1,2,3,4,5\}

and

f(a)=1

for all

f(a),a∈ A

Explanation

Which of these is a valid permutation of the set

S=\{1,2,3,4,5,6,7,8\}

(4,5,3,2,6,1,7,8,3)

Explanation

(8,4,6,1,3)

Explanation

(1,2,3,4,5,6,7,8)

Explanation

d: none of the above Explanation

Explanation

What is the order of the permutation

(1,3,2)(4,5)

a: 3 Explanation

Explanation

b: 2 Explanation

Explanation

c: 6 Explanation

Explanation

d: 5 Explanation

Explanation

How many 2-cycles are needed to represent the permutation

(1,4,3,2)

a: 2 Explanation

Explanation

b: 3 Explanation

Explanation

c: 4 Explanation

Explanation

d: 5 Explanation

Explanation

If a permutation has cycle structure

(1,2)(3,4,5)(6,7,8,9)

, what is its order?

a: 12 Explanation

Explanation

b: 24 Explanation

Explanation

c: 4 Explanation

Explanation

d: 6 Explanation

Explanation

Given permutations

α=(1,2,3)

and

β=(2,3,4)

in cycle notation, what is

α∘β

(apply

β

first, then

α

(1,2)(3,4)

Explanation

(1,2,4,3)

Explanation

(1,3)(2,4)

Explanation

(1,2,3,4)

Explanation

What is the sign (parity) of the permutation

(1,2,3,4,5)(6,7)

a: Even Explanation

Explanation

b: Odd Explanation

Explanation

c: Cannot be determined Explanation

Explanation

d: Always positive Explanation

Explanation

In the 15-puzzle, why are some initial configurations impossible to solve?

a: Because the grid is too small Explanation

Explanation

b: Because the empty space is in the wrong position Explanation

Explanation

c: Because of the parity of the permutation Explanation

Explanation

d: Because some numbers are too far from their target position Explanation

Explanation