Question

Suppose sets \(A\) and \(B\) are in a universal set \(U\). Draw Venn diagrams for \(\overline{A \cap B}\) and \(\bar{A} \cup \bar{B}\). Based on your drawings, do you think it's true that \(\overline{A \cap B}=\bar{A} \cup \bar{B}\) ?

Short answer

Based on the Venn Diagrams, the statement that \(\overline{A \cap B} = \overline{A} \cup \overline{B}\) is not true. The first set is the set of all elements not common to A and B, while the second one includes all elements outside each of the sets A and B, but includes those that are in either A or B but not both.

Step-by-step solution

Draw Venn Diagram for \(\overline{A \cap B}\)

Start by drawing circles representing sets 'A' and 'B' overlapping in a rectangle which represents the Universal Set 'U'. All the areas inside the circles represent elements that belong to those sets, and the overlapping part between sets A and B represents the intersection of these two sets (i.e., elements common to both sets). The symbol \(\overline{A \cap B}\) represents the complement of \(A \cap B\), which means all the elements not in \(A \cap B\). This includes all elements outside the overlapping region of set A and B.

Draw Venn Diagram for \(\overline{A} \cup \overline{B}\)

The symbols \(\overline{A}\) and \(\overline{B}\) represent the complements of Set A, and Set B, respectively which means all the elements that are not in Set A and Set B. The symbol \(\overline{A} \cup \overline{B}\) represents the union of the complements of Sets A and B. In the Venn Diagram this includes all areas not in A (outside the circle A) and not in B (outside the circle B). This is the union of separate areas outside each circle.

Compare the Two Diagrams

Compare the representations of \(\overline{A \cap B}\) and \(\overline{A} \cup \overline{B}\). If the two diagrams are entirely equivalent (i.e., they show exactly the same areas) then it can be stated that \(\overline{A \cap B} = \overline{A} \cup \overline{B}\) is true. Otherwise, it is not true.