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Sets and Set Operations

Class Note 04:

Sets and Set Operations

Computer Sci & Eng Dept

SUNY Buffalo

c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 45

Sets

Definition:

A Set is a collection of objects that do NOT have an order.

Each object is called an element.

We write e S if e is an element of S; and e S if e is not an

element of S.

Set is a very basic concept used in all branches of mathematics and

computer science.

How to describe a set:

Either we list all elements in it, e.g., {1, 2, 3}.

Or we specify what kind of elements are in it, e.g.,

{a | a > 2, a R}.

(Here R denotes the set of all real numbers).

c Xin He (University at Buffalo) CSE 191 Discrete Structures 3 / 45

Example sets

N = {0, 1, 2, . . .}: the set of natural numbers.

(Note: in some books, 0 is not considered a member of N.)

Z = {0, -1, 1, -2, 2, . . .}: the set of integers.

Z+ = {1, 2, 3, . . .}: the set of positive integers.

Q = {p/q | p Z, q Z, q = 0}: the set of rational numbers.

Q+ = {x | x Q, x > 0}: the set of positive rational numbers.

R: the set of real numbers.

R+ = {x | x R, x > 0}: the set of positive real numbers.

Definition:

The empty set, denoted by , is the set that contains no elements.

c Xin He (University at Buffalo) CSE 191 Discrete Structures 4 / 45

More example sets

A={Orange, Apple, Banana} is a set containing the names of

three fruits.

B={Red, Blue, Black, White, Grey} is a set containing five colors.

{x | x takes CSE191 at UB in Spring 2014} is a set of 220

students.

{N,Z,Q,R} is a set containing four sets.

{x | x {1, 2, 3} and x > 1 } is a set of two numbers.

Note: When discussing sets, there is a universal set U involved, which

contains all objects under consideration. For example: for A, the

universal set might be the set of names of all fruits. for B, the universal

set might be the set of all colors.

In many cases, the universal set is implicit and omitted from

discussion. In some cases, we have to make the universal set explicit.

c Xin He (University at Buffalo) CSE 191 Discrete Structures 5 / 45

Equal sets

Definition:

Two sets are equal if and only if they have the same elements.

Note that the order of elements is not a concern since sets do not

specify orders of elements.

We write A = B, if A and B are equal sets.

Example:

{1,2,3} = {2,1,3}

{1, 2, 3, 4} = {x Z and 1 x < 5}

c Xin He (University at Buffalo) CSE 191 Discrete Structures 6 / 45

Subset

Definition:

A set A is a subset of B if every element of A is also in B.

We write A B if A is a subset of B.

Clearly, for any set A, the empty set (which does not contain any

element) and A itself are both subsets of A.

Definition:

If A B but A = B, then A is a proper subset of B, and we write A B.

Fact:

Suppose A and B are sets. Then A = B if and only if A B and B A.

This fact is often used to prove set identities.

c Xin He (University at Buffalo) CSE 191 Discrete Structures 7 / 45

How to use the point of intersection calculator? Graph the functions in a viewing window that contains the point of intersection of the functions. Press [2nd] [TRACE] to access the Calculate menu. Press [5] to select the intersect option. Select the first function. ... Select the second function. ... Use the right- and left-arrow keys to move the cursor as close to the point of intersection as possible. ... More items...

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