Main Concept

Venn diagrams are often used to show logical relationships between a finite number of sets. In a Venn diagram, the outer rectangle represents the universal set, U  (the set of all possible elements), while the enclosed circles represent distinct sets of elements. These sets can also be denoted using uppercase letters A, B, C, etc.

Using the first diagram below, we can illustrate the fundamental properties of a set:

A refers to a set of elements which share some defining characteristic, and can be illustrated by shading the interior of the circle labeled A.

$A^C$, the complement of $A$, refers to the set containing all elements in the universal set which do not belong to A. This can be illustrated by shading the area within the rectangle, but outside of the circle labeled A.

Using the second diagram, we can illustrate the relationships which exist between two sets:

The intersection of sets B and C, denoted $B \bigcap C$, refers to the set containing only elements which belong to both B and C. This can be illustrated by shading the intersection of the two circles labeled B and C.

Two sets are said to be disjoint if they have no elements in common and so their intersection is empty, denoted $B \bigcap C \=\varnothing$.

The union of sets B and C, denoted $B \bigcup C$, refers to the set containing all distinct elements which belong to $B$ or $C$. This can be illustrated by fully shading both circles labeled B and C, including their intersection (since all elements of the intersection belong to both B and C, they need to only be listed once when describing the elements of the union).

You will now notice that in the first diagram, the union of A and $A^C$ contains the entire universal set U. So, we can conclude:   $A \bigcup A^C \= U$.

Also, since there are no elements which exist in both A and $A^C$, we can conclude that these two sets are disjoint and so $A \bigcap A^C \= \varnothing$.

Also, in the second diagram, you will notice that the intersection of $B^C$ and $C^C$ creates the same shading pattern as the complement of $B \bigcup C$, while the union of $B^C$ and $C^C$ creates the same shading pattern as the complement of $B \bigcap C$.

These properties are known as DeMorgan's Laws: ${\left(B \bigcup C\right)}^C \= B^C \bigcap C^C$ and ${\left(B \bigcap C\right)}^C \= B^C \bigcup C^C$ .

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