Chapter 1 SETS

Question 1

Set

Answer 1

Set – A set is a collection of objects

Example: J = {2,4,1,3,7}

Question 2

Order Set

Answer 2

Order Set – A ordered set is a collection objects in an order

Example: J = {1,2,3,4,5}

Question 3

Elements

Answer 3

Elements – The objects that make up a set are called its elements

• If a is an element of a set A, then we write a ∈ A
• If a is not an element of a set A, then we write a ∈/ A

Ex.1 The set S = {1, 2, 3, 4, 5} has 5 elements. Also it can be written as S = {2, 3, 4, 5, 1}. (Note that the order in which the elements are listed does not matter.)

Ex 2 The set of all positive even integers less than 10 is S = {2, 4, 6, 8}. In this case, 2 is an element of S, or 2 ∈ S. Also 3 ∈/ S

Question 4

Empty Set

Answer 4

Empty Set - The set which contains no elements is called the empty set and denoted by ∅.

Question 5

Cardinality

Answer 5

Cardinality - The number of elements of a set S is called cardinality of S and denoted by |S|.

Question 6

Finite(in terms of sets)

Answer 6

Finite - A set S is finite if |S| = n for some nonnegative integer n.

Question 7

Infinite(in terms of sets)

Answer 7

Infinite - A Set S is infinite if it is not finite and denoted by |S| = ∞.

Question 8

Subset

Answer 8

Subset - A Set A is called a subset of B if every element of A also belongs to B. In this case, we write A ⊆ B.

Question 9

Universal Set

Answer 9

Universal Set - In a typical discussion of sets, we are ordinarily concerned with subsets of some specific set U, called the universal set.

Question 10

Equal (in terms of sets)

Answer 10

Equal (in terms of sets) - Two Sets A and B are said to be equal if they have exactly the same elements. In this case, we write A = B. In other words, A = B if and only if A ⊆ B and B ⊆ A.

Question 11

Proper Subset

Answer 11

Proper Subset- If A is a subset of B but A does not quals B, then A is a proper is a proper subset of B

Question 12

Proposition of Subsets

Answer 12

Proposition of Subsets

Let A, B, and C be sets.

(1) A ⊆ A
(2) If A ⊆ B and B ⊆ C, then A ⊆ C.
(3) ∅ ⊆ A.

Question 13

Power Set

Answer 13

Power Set - the set of all subsets of a set A is called a power set of A and denoted by P(A) where A is the set.

the cardinailty of the power set is the number of 2 ^ number of elements in the set A

Question 14

Intersection

Answer 14

Intersection - The intersection of A and B, denoted by A ∩ B, is the set of all elements belonging to both A and B.

Question 15

Disjoint

Answer 15

Disjoint - Two Sets A and B are said to be disjoint if A ∩ B = ∅

Question 16

Union

Answer 16

Union - The union of two sets A and B, denoted by A ∪ B, is the set of all elements belong to A or B.

Question 17

Difference

Answer 17

Difference - Let A and B be sets. The difference of two sets A and B is

A − B = {x | x ∈ A and x ∈/ B}.

It is also written as A \ B.

Question 18

Complement

Answer 18

Complement- Let U be the universal set that is all sets being discussed are subsets of U.For set A, the difference of U - A is called the complement of A and denoted A(over line on top)

Question 19

Pairwise Disjoint

Answer 19

Pairwise Disjoint- A Collection S of subsets of a set A is called pairwise disjoint if every two distinct subsets that belong to S are disjoint.

Example: Let A = {1, 2, 3, 4, 5, 6} and S the collection of nonempty subsets of A:

S = {{1, 3, 6}, {2, 4}, {5}}

Question 20

Partition

Answer 20

Partition - Let A be a set. A partition of A is a collection S of nonempty pairwise disjoint subsets of A whose union is A.

Question 21

Cartesian product

Answer 21

Cartesian product - Let A and B be sets. The Cartesian product A × B is the set of all ordered pairs

A × B = {(a, b) | a ∈ A, and b ∈ B}.

Question 22

Union & Interception of Sets

Answer 22

Question 23

Index Sets & Index Collection of Sets

Answer 23