Elementary Set Theory

Introduction to elementary set theory:

A set theory is a collection of elements or the items can be considered as a whole. If the set contains only a few items or elements, then the set can be defined by listing them in braces. For example: A = {1,2,3}. Now we are going to see about the elementary set theory.

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Elementary set theory:


Now we are going to see about the elementary set theory as follows,

Set and Sample Space:

The sets are the important part and the basic concepts in mathematics and probability. Sets are usually consists of collection of some elements.

Example of sample spaces:

Pick a card from a pack of 52 cards:

Sample space = {1, 2 ...52}, which is a infinite sample space.

Subset:

A subset is can be defined by some property of its elements.

For example, let P = {1,2,3,4}, and let Q = {2,}. Then Q can be defined as the set of all elements of P which are even, or in symbols:

Q ={x  `in` A |  x is even}.

The operation properties are given as follows,

The intersection operation has the properties are given as,

Commutative: A `nn` B = B `nn` A.

Associative: (A `nn` B) `nn` C = A `nn` (B `nn` C).

The union operation has several properties

Commutative: A `uu` B = B `uu` A.

Associative: (A `uu` B) `uu` C = A `uu` (B `uu`C).

Ordered pairs:

An ordered pair contains set of two elements that are arranged in a specified order. An ordered pair is usually written as (x, y).

Relations:

A relation R is on a set A, where A is simply a set of ordered pair of element of A.

Functions:

A function ‘f’ from the set X to the set Y is a rule given any element x of A, This concept is often expressed symbolically as f: X--->Y

Problems for elementary set theory:


Example 1:

A = {1, 2, 3, 4, 5, 6, 7, 8, 11, 12} B = {9, 10, 11, 12, 13). Determine the union and intersection of set A and B.

Solution:

The union of set is nothing but including both the set elements in a single set

(B `uu` C) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, (B ∩ C) = {11, 12}.

Example 2:

P = {f, g, h, i, j, k, l} Q = {g, j, l, m, n}. Determine (P – Q) and compliment of C related to Q.

Solution:

The difference of the two sets P and Q are given as,

(P – Q) = {f, h, I, k}, Cc = {m, n}