Chapter 3 : Relations and Functions

Question 1

What is the most direct way to express a relationship between elements of 2 sets?

Answer 1

1. Use Ordered Pairs

Question 2

Ordered pairs made up of 2 related elements, for this reason, it is also called as what?

Answer 2

1. Binary Relations

Question 3

What is the term Applications in Relations and Functions?

Answer 3

1. Methematical Concepts related to Relations and Functions used in the real world

Question 4

What relationship / connections that Applications explain?

Answer 4

1. A program and a variable it uses
2. A computer language and a valid statement in the language

Question 5

What structure does elements of sets are represented in?

Answer 5

1. Relation

A = { students }
B = { courses }
R ( Relation ) = ( a , b )
= ( students , courses )

( Jason, MCFC ) , ( Jason, IOOP )

Question 6

How is Ordered Pairs written?

Answer 6

1. Written as ( a , b ) where a is the first element and b is the second element

Question 7

What does aRb represent?

Answer 7

1. ( a, b ) ∈ R

• R - Relation
• a is said to be related to b by R

Question 8

Is { a , b } = { b , a }?

Answer 8

1. True

Question 9

Is ( a , b ) = ( b , a )?

Answer 9

1. False

( a , b ) ≠ ( b , a )

Question 10

What is the purpose for Relations and Cartesian Product Set?

Answer 10

1. Relation - to express and test if a relation holds between two individual elements
2. Cartesian Product Set - combine two sets to create a new set of all possible pairings

• Example
  A = {flour, sugar, eggs}
  B = {bake, fry}

Relation ( Is like asking a question like “Is it delicious to bake flour?” )
- (flour)R(bake) would be FALSE (just baked flour isn’t delicious)

Cartesian Product Set
A × B = {(flour, bake), (flour, fry), (sugar, bake), (sugar, fry), (eggs, bake), (eggs, fry)}

Question 11

What does A x B means?
( Both A and B are sets )

Answer 11

1. A crosses B
2. Cartesian Product of A and B

• A x B = { ( a , b ) | a ∈ A , b ∈ B }

Question 12

What can A x A be represented as?

Answer 12

1. A²

• A x A = { ( a₁ , a₂ ) | a₁ ∈ A , a₂ ∈ B }

Question 13

Given
A = { 1 , 2 }
B = { p , q }
A x B = { ( 1 , p ),( 1 , q ), ( 2 , p ), ( 2 , q ) }

Determine True / False
1. R₁ = { ( 1 , p ) }
2. R₂ = { ( 2, k ) }
3. R₃ = { ( 1 , q ) , ( 2 , p ) }

Answer 13

1. True
2. False ( Not the relations from A to B )
3. True

Question 14

How to represent relations? ( 2 )

Answer 14

1. Arrow Diagram
2. Table

Question 15

What is Domain of R?

Answer 15

1. The set of all elements in A that are related to some element in B

Question 16

What is the representation of Domain of R?

Answer 16

1. Dom( R )

Question 17

What is Range of R?

Answer 17

1. The set of all elements in B that are related to some element in A

Question 18

What is the representation of Range of R?

Answer 18

1. Ran( R )

Question 19

A = { Alice , Bob }
B = { CS101, CS102 }

R = { ( Alice , CS101 ) , ( Alice , CS102 ) , ( Bob , CS101 ) , ( Bob , CS102 ) }

Can you list out which is Domain and which is Relation?

Answer 19

1. Domain = { Alice, Bob }
2. Range = { CS101 , CS102}

Question 20

How many elements that a relation on set A is a subset of A x A ?

Answer 20

1. 2^n^2

• Example
  A = { a , b ,c }

The number of possible relations on set A = 2^3^2 ( 512 )

3 = { a , b , c }
2 = A x A ( 2 Sets )

Question 21

What is an equivalence relation?

Answer 21

1. It is a binary relation that is
   
   • Reflective
   • Symmmetric
   • Transitive

Question 22

Let R be a relation of set A, how to determine that R is reflextive?

Answer 22

1, if ( a , a ) ∈ R for every element a ∈ A

( a , a )
* ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 )

Question 23

Let R be a relation of set A, how to determine that R is symmetric?

Answer 23

1. If ( b , a ) ∈ R whenever ( a , b ) ∈ R, for some a , b ∈ A

( b , a ) must have ( a , b )
( 1 , 2 ) must have ( 2 , 1 )
( 3 , 4 ) must have ( 4 , 3 )

Question 24

Let R be a relation of set A, how to determine that R is transitive?

Answer 24

1. If ( a, b ) ∈ R and ( b , c ) ∈ R then ( a , c ) ∈ R , for a,b,c ∈ A

( a , b ) and ( b , c ) must have ( a , c )
( 2 , 4 ) and ( 4 , 1 ) must have ( 2 , 1 )

Question 25

Functions are also called as what? ( 2 )

Answer 25

1. Mappings 2. Transformation

Question 26

What is a under f called when b = f ( a ) ?

Answer 26

1. Image

Question 27

What is a called?

Answer 27

1. The object of b

Question 28

What is f called when f ( A ) = { f(a) | a ∈ A }?

Answer 28

1. The range of f

Question 29

What is the noun called for a of f?

Answer 29

1. Domain

Question 30

What is the noun called for b of f?

Answer 30

1. Range

Question 31

How to write function?

Answer 31

1. f : A -> B

Question 32

What is the rule for function? ( 2 )

Answer 32

1. One to One 2. Many to One 3. Elements inside A must have a relation Not 1. Many to Many 2. One to Many

Question 33

What is the difference between Relation and Function?

Answer 33

1. Function yields a **single result** for any element in its domain 2. Relation allows **multiple mappings** between the domain and the co-domain * Function - Age, Square Root Relation - Students enrolled in one course

Question 34

Let f be the function defined by the rule f ( x ) = X^2 f 0 0 1 1 2 2 3 4 Define Domain & Codomain and list out all outcomes

Answer 34

1. Domain = { 0 , 1 ,2 } 2. Codomain = { 0 , 1 , 2 , 3 ,4 , } 3. f ( x ) = X^2 f ( 0 ) = 0 f ( 1 ) = 1 f ( 2 ) = 4

Question 35

Determine 1. ( f + g ) ( x ) = 2. ( f - g ) ( x ) = 3. ( fg ) ( x ) = 4. ( f/g ) ( x ) =

Answer 35

1. f ( x ) + g ( x ) 2. f ( x ) - g ( x ) 3. f ( x ) . g ( x ) 4. f ( x ) / g ( x )

Question 36

Determine ( Composite Functions ) 1. g o f ( x ) 2. f o g ( x )

Answer 36

1. g [ f ( x ) ] 1. f [ g ( x ) ]

Question 37

When is f^-1 ( b ) = a ? ( Inverse Function )

Answer 37

1. When f ( a ) = b

Question 38

Given that f ( x ) = 2x + 1, find f^-1 ( x ) and f^-1 ( 15 )

Answer 38

Let y = f ( x ) f^-1 ( y ) = x y = 2x +1 2x = y - 1 x = ( y - 1 ) / 2 f^-1 ( y ) = ( y - 1 ) / 2 f^-1 ( x ) = ( y - 1 ) / 2 f^-1 ( 15 ) = ( 15 - 1 ) / 2 = 7