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# Union of Sets

Top
 Sub Topics A fundamental operation through which sets are combined and related to each other. The term union tells us it is a collection of sets containing distinct elements. It is denoted by $\cup$. Combining the members of each we can find the union of any number of sets. For two sets A and B it is pronounced as A union B and is denoted as A $\cup$ B. In A $\cup$ B the set contains elements that belong to either A or to B or to both. Union of sets will take everything that is in either of the sets. If we are doing any experiment then we find some results or outcomes of that experiment then the collection of these outcomes is called event simply, any subset of the sample space is an event.Consider an experiment, a fair die is rolled three times and its outcomes are 1, 2, 5 respectively then its outcomes can be represented as an events as following E = {1, 2, 5}In this way, we can say that E is an event of rolling a die three times.Since, event is a subset of any sample space then this subset consists of elements. On this basis there are two types of the events. i) Any event that is of a single result or any it contains a one and only outcome then this is said elementary event or simple event. For Example: E = {x} ii) Event which consist of two or more than two results then we say that this event is compound event. For Example: E ={1, 3, 6} and E ={x, y, z}

## Definition

Union of sets is simply a union of two or more than two sets. If two sets representing events are combined then we form a single set representing another event that contains all the events in a single set then it is said to be a union of sets. The union of sets is denoted by sign $\cup$. If A and B are two events then A $\cup$ B is called union of A and B.

Suppose that two sets are given A and B then
The union of two sets A and B is the set which consists all the elements of A and B:
A = {a, b, c} and B = {x, y, z}
then  A $\cup$ B = {a, b, c, x, y, z}
The venn diagram is as following:

There is an another example   A = {0, a, b, c, 1, 2} and , B={1, 2}
A $\cup$ B={0, 1, 2, a, b, c,}

Remember that element is written only once even if they exist in both the sets.

## How to Calculate

The calculation of Union of sets is easy to understand. If we have two sets and we are asked to find the union then we just collect all the elements of both given sets and put these in a new set and this new formed set.

Like We have two sets A={1, 2, 3} and B={4, 6} these events are combined then we get A U B = {1, 2, 3, 4, 6}

Another Example: P = { x, y, z} and Q = {x, a, b} then P U Q = {x, y, z, a, b}

In above example we have x in both P and Q events but P U Q doesn't have two x because we know any element doesn't repeat itself in set.

## Examples

Example 1: If P={A, B, C} and Q = {1, 2, 3} then find out union of P and Q.

Solution: Given sets or events P={A, B, C} and Q={1, 2, 3}
So,
P U Q= {1, 2, 3, A, B, C}

Example 2: If P={x, y}, Q={a, b, c,} and R = {1, 2}}then find out P U Q U R.

Solution:
Given sets are
P={x, y}, Q={a, b, c,} and R = {1, 2}
So,
P U Q= {x, y, a, b, c, 1, 2}

Example 3: If A = { All positive numbers} and B = { All negative numbers} then find out A U B.

Solution:   Given events

A = { All positive numbers} and B = { All negative numbers}

Then A U B = {All integers}