3.3 Union of sets

We are familiar with basic algebraic operations. These basic mathematical operations, however, are not valid in all contexts. For example, algebraic operation such as addition has different details, when operated on vectors. Clearly, we expect that these operations will also be not same in the case of sets – which are collections and not individual elements.

Nevertheless, set operations bear resemblance to algebraic operation. For example, when we combine (not add) two sets, then the operation involved is called “union”. We can see that there is resemblance of the intent of addition, subtraction etc in the case of sets also.

Venn diagrams

Venn diagrams are pictorial representation of sets/subsets and relationship that the sets/subsets have among them. It helps us to analyze relationship and carry out valid set operations in a relatively easier manner vis – a – vis symbolic representation.

Universal set

Universal set is the largest set among collection of sets. Importantly, it is not the collection of everything as might be conjectured by the nomenclature. For example, "R", is universal set comprising of all real numbers. The rational numbers, integers and natural numbers are its subset. In other consideration, we can call integers as universal set. In that case, sets such as {1,2,3}, prime numbers, even numbers, odd numbers are subset of the universal set of integers.

The universal set is pictorially represented by a region enclosed within a rectangle on Venn diagram. For illustration, consider the universal set of English alphabets and universal set of first 10 natural numbers as shown in the top row of the figure

Many times, however, we may not be required to list elements of a universal set. In such case, we represent the universal set simply by a rectangle and the symbol for universal set, “U”, in the corner. This is particularly helpful, where number of elements in universal set are very large.

The subsets of the universal set are represented by closed curves – usually circles. The subset of vowels (V) is shown here within the circle with the listing of elements. Note that we have not listed all the alphabets for universal set and used the symbol “U” in the corner only.

Union of sets

Union works on two operands, each of which is a set. The operation is denoted by symbol " $\cup$ ". Now, the question is : what do we expect when two sets are combined? Clearly, we need to enlist all the elements of two sets in the resulting set.

Union of two sets

The union of sets “A” and “B” is a third set, which consists all the elements of two sets.

In symbol,

$$A\cup B=\{x:x\in Aorx\in B\}$$

The word “or” in the set builder form defining union is important. It means that the element “x” belongs to either “A” or “B”. The element may belong to both sets (common to two sets), but not necessarily. We can, therefore, infer that union set consists of :