Set Difference

Today, we delve into an integral concept in set theory—the difference between sets—and discuss how this operation is performed.

The difference between two sets, referred to as A and B, is simply a set comprising elements that exist in set A but do not belong to set B. Represented by the minus symbol, "-," we write this as $$ A-B$$

In essence, the set difference, A-B, includes all elements originating from set A that are absent in set B.

Mathematically, this is expressed as:

$$ A - B = \{x | x ∈ A \ , \ x ∉ B \} $$

You can think of this as a form of subtraction operation applied to the elements of the two sets.

When calculating the difference between sets A and B, the method involves comparing the two sets and removing any common elements from set A.

You may find Venn diagrams a useful tool in visually representing the difference between two sets. In such diagrams, the set difference A - B corresponds to the section of set A shaded in gray, exclusive of its intersection with set B.

From a geometric perspective, the set difference A-B represents the area of set A (shaded gray) that does not intersect with set B.

Let's ground this theory with a practical example.

Suppose we have two finite sets, A and B.

$$ A = \{2, 4, 6, 7, 8 \} $$

$$ B= \{1, 3, 8, 7 \} $$

Here, the set difference A - B is the set {2, 4, 6}, which includes all elements that are found in set A but not in set B.

$$ A - B = \{2, 4, 6, \color{red}7, \color{red}8 \} - \{1, 3, \color{red}8, \color{red}7 \} $$ $$ A - B = \{2, 4, 6 \}$$

In this instance, the numbers 7 and 8 are common to both sets. To derive the set difference A-B, we remove these common elements from set A.

To further enhance your understanding, I recommend drawing the sets using Euler-Venn diagrams.

 

Dal punto di vista geometrico l'insieme differenza A-B è l'area (grigia) dell'insieme A che non è sovrapposta all'insieme B.