Lookup for a Set should be O(1). This is true for sets and hashtables (eg maps) in any language.
The way this is possible is that sets store values differently than arrays.
In an array values are stored sequentially based on what their place is in the array and where that array is in memory, so to find your item you need to scan through the array sequentially to find your item (unless it's a sorted array, then you can use binary search at O(logn)).
Sets declare a block of memory, like an array, but instead of putting items in memory in sequence, like an array, they determine the index of the item to add by running the item through a hash function (essentially a function that takes in an object and returns an evenly distributed, very large random number), and then modulousing the result of that hash function by the size of the memory block they have.
So, when you call contains($needle, $mySetHaystack), php will take $needle, and feed it into a hashfunction, which will return a big number like 9283472378, then it takes the length of $mySetHaystack (let's say 31), and does 9283472378 % 31 = 28, so it checks the 28th index of $mySetHaystack to see if $needle is there. Everything in this list of operations is independent of the size of $mySetHaystack, hence the perf being O(1).
If a hash function returns the same value for two different items (a hash collision, which totally happens), or if the modulo of that value is the same, then an array of values is stored in the set at that index. Since sets don't allow duplicate values, this happens rarely and is negligible from a perf perspective.
You should check out the wikipedia page on hash tables (similar to sets), as there are lots of pictures that will make this concept easier to understand.
