1.3.7.2 Intersection of Sets

The intersection of two sets is the set containing all the elements that are common to both sets. Mathematically, the intersection of sets  and  is denoted as   . A key characteristic of the intersection is its specificity. It includes only those elements that are present in both sets. If an element is in one set but not in the other, it is not included in the intersection. Another important aspect is that each element in the intersection is unique and shared by both sets.

In set builder form, the intersection of sets  and  is represented as:

This can be interpreted as “The intersection of sets  and , denoted by , consists of all elements  such that  is an element of set  and  is an element of set .” In other words, it includes only those elements that are members of both set  and set . For instance, if  = {1, 2} and  = {2, 3}, then:

Example 1.19. Referring back to Example 1.18 from the previous section, the intersection of the sets representing the 'Science' and 'Environmental Studies' departments would be the set containing the subject(s) common to both, as follows:

        

Fig 1.7 displays the Venn diagram of the intersection set, represented by   . As the intersection encompasses all elements present in both sets  and , the new set (highlighted in the blue region) includes the element 'Biology'.

If  and  have no elements in common, then  and  are said to be disjoint sets and their intersection is an empty set. In symbols,    = . For example, let  = {1, 2, 3, 4} and  = {5, 6, 7, 8}. The intersection of  and , denoted by   , is .

Example 1.20. The following table demonstrates the union and intersection of sets for various examples:

Example	Sets	Union	Intersection
Two Sets with Common Elements	A = {1, 2, 3}, B = {3, 4, 5}	{1, 2, 3, 4, 5}	{3}
Two Sets with No Common Elements (Disjoint Sets)	C = {a, b, c}, D = {x, y, z}	{a, b, c, x, y, z}	{}
Multiple Sets	E = {1, 2, 3}, F = {2, 3, 4}, G = {3, 5}	{1, 2, 3, 4, 5}	{3}
Empty Set	H = {apple, orange}, I = {}	{apple, orange}	{}
Sets with Nested Elements (Sets within Sets)	J = {1, {2, 3}, 4}, K = {{2, 3}, 5}	{1, {2, 3}, 4, 5}	{{2, 3}}
Universal Set	L = {red, green, blue}, U = {red, green, blue, yellow, white}	{red, green, blue, yellow, white}	{red, green, blue}