Chapter 4 notes

Question 1

What is the basic definition of Relations between sets when: Suppose A and B are sets. then A is called a subset of B: A subset B

Answer 1

Suppose A and B are sets. then A is called a subset of B: A subset B

iff every element of A is also an element of B

Example: A subset B, For all x , if x E A then x E B

Example 2: A not subset B, There exists x such that x E A and x E! B

Question 2

What is the definition when A and B are sets and A = B?

Answer 2

This is true iff every element of A is in B and every element of B is in A

Question 3

What is the definition of Union of A and B ( A U B)

Answer 3

Normally U stands for Universal set.

x exists U | x exists A or x exists B

The U is essentially an ‘or’ statement. Looks like up facing horseshoe

Question 4

What is the definition of the Intersection of A and B

Answer 4

x exists U | x exists A and x exists B

This is essentially an and statement. Looks like down facing horseshoe

Question 5

What is the definition of: Difference of set B minus A?

Answer 5

x exists U | x exists B and x doesn’t exist A

Question 6

What is the definition of the Complement of set A, A^c

Answer 6

x exists U | x doesn’t exist A

Question 7

What is A intersection B a subset of?

Answer 7

This is a theorem and is always a subset of A

Question 8

What is A always a subset of?

Answer 8

A is always a subset of A union B (theorem)

Question 9

If A subset B and B subset C, then _____?

Answer 9

A subset C (theorem)

Question 10

What is the distributive law in Set Theory

Answer 10

For any sets A,B and C:

A union (B inter C)= (A union B) inter (A union C)

Question 11

What are the commutative Laws in Set Theory

Answer 11

A inter. B = B inter A

A union b = B union A

Question 12

What are the Associative Laws in Set Theory

Answer 12

( A inter B) intersection C = A inter (B inter C)

A union B) union C = A union (B union C

Question 13

What are the distributive Laws in Set Theory

Answer 13

A union (B inter. C) = (A union B) inter. (A union C)

A inter. ( B union C) = (A inter. B) union ( A inter. C)

Question 14

What is the double Complement Law in Set Theory

Answer 14

(A^c)^c = A

Question 15

What is De Morgan’s Law in Set Theory

Answer 15

( A inter B)^c = A^c union B^c

(A union B)^c = A^c inter B^c

Question 16

What is the absorption Law in Set Theory

Answer 16

A union ( A inter B) = A

A inter (A union B) = A

Question 17

What is an Empty Set

Answer 17

The unique set with no elements is called the Empty set and denoted by 0 with a slash through it.

For all sets A

1. 0 is a subset of A
2. A union 0 = A
3. A inter 0 = 0
4. A inter A^c = 0

Question 18

When are A and B called disjointed sets

Answer 18

They are disjointed iff A inter B = 0

ex. A = (1,2) B= (3,4)

Question 19

When are sets A1, A2,…An considered mutually disjoint?

Answer 19

IFF for all sets Ai inter Aj = o whenever i =! j

ex. A = (1,4) B=(2,5), C=(3)

Question 20

What is a Partition in set theory

Answer 20

A collection of nonempty sets.

A1,A2,…,An are mutually disjoint

Question 21

What is a Power Set

Answer 21

Given a set A, the power set of A, denoted P(A) is the set of all subsets of A.

Ex. P(a,b) = (0, a, b, (a,b))

Properties:
If A subset B then P(A) subset P(B)
If a set A has n elements then P(A) has 2^n elements

Question 22

What is the Cartesian product of 2 sets A and B

Answer 22

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

Question 23

What is the inclusion-exclusion principle of a Set

Answer 23

|A union B| = |A| + |B| - | A int. B|

Ex. A, B, C
|A union B union C| = |A| + |B| +|C| - |A int B| - |A int C| - |B int C| + |A int B int C|

Question 24

What does |A| indicate

Answer 24

The number of elements in the set

Question 25

What is the definition of Function pertaining to Set theory

Answer 25

a relationship between elements of 2 sets such that no element of the first set is related to more than once element of the second set.

Question 26

What is the Domain of sets

Answer 26

the set which contains the values to which the function is applied.

Question 27

What is the Co-domain of a sets

Answer 27

the set which contains the possible values (results) of the function

Question 28

What is the Range of Sets

Answer 28

the set of actual values produced when applying the function to the values of the domain.

Question 29

When talking about functions in Set theory define the following:

• f
• X
• Y
• x exists X y exists Y
• f(x) = y

Answer 29

• the function name
• the domain
• the co-domain
• f sends x to y
• f of x; the value of f at x; the image of x under f