A set is a simply a collection of values without any particular order. It can contain any data type and we can run different set operations such as union, intersection, complement, and so on. As a set only contains values, we can construct a basic PHP array and assign values to it so that it grows dynamically. The following example shows two sets that we have defined; one contains some odd numbers and the other one has some prime numbers:

$odd = []; 
$odd[] = 1; 
$odd[] = 3; 
$odd[] = 5; 
$odd[] = 7; 
$odd[] = 9; 

$prime = []; 
$prime[] = 2; 
$prime[] = 3; 
$prime[] = 5; 

In order to check the existence of a value inside the set along with union, intersection, and complement operation, we can use the following example:

if (in_array(2, $prime)) { 
    echo "2 is a prime"; 
} 

$union = array_merge($prime, $odd); 
$intersection = array_intersect($prime, $odd); 
$compliment = array_diff($prime, $odd);  

PHP has many built-in functions for such operations and we can utilize them for our set operations. But we have to consider one fact: since the set is not ordered in any particular way, searching using the in_array() function might have the complexity of O(n) in worst case scenario. Same goes for the array_merge() function, as it will check each value from one array with another array. In order to speed things up, we can modify our code a little bit and make it efficient:

$odd = []; 
$odd[1] = true; 
$odd[3] = true; 
$odd[5] = true; 
$odd[7] = true; 
$odd[9] = true; 

$prime = []; 
$prime[2] = true; 
$prime[3] = true; 
$prime[5] = true; 

if (isset($prime[2])) { 
    echo "2 is a prime"; 
} 

$union = $prime + $odd; 
$intersection = array_intersect_key($prime, $odd); 
$compliment = array_diff_key($prime, $odd); 

If we analyze this code, we can see that we are using an index or key to define the set. Since a PHP array index or key lookup have a complexity of O(1), it makes the searching much faster. As a result, all the lookup, union, intersect, and complement operations will take lesser time compared to the previous example.