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Chapter 7: Problem 55

Use Venn diagrams to illustrate each statement. $$ A \cup(B \cup C)=(A \cup B) \cup C $$

Short Answer

Expert verified
First, draw separate Venn diagrams for the LHS and RHS of the equation, i.e., A 鈭� (B 鈭� C) and (A 鈭� B) 鈭� C. In each diagram, shade the region corresponding to the Union of the sets involved. Then, compare both Venn diagrams and observe that both yield the same outcome, illustrating that the given statement is true: \[ A \cup(B \cup C)=(A \cup B) \cup C \]

Step by step solution

01

Draw Venn diagrams for LHS and RHS of the equation separately.

First, we will draw a Venn diagram for the left-hand side of the equation, i.e., A 鈭� (B 鈭� C). To do this, we need to first find the Union of sets B and C, and then find the Union of set A with the result of B 鈭� C. Similarly, for the right-hand side of the equation, i.e., (A 鈭� B) 鈭� C, we first need to find the Union of sets A and B, and then find the Union of the result with set C.
02

Illustrate the Union of B and C.

Draw three circles representing sets A, B, and C and label them accordingly. Then, shade the region corresponding to the Union of sets B and C (B 鈭� C), which includes all the elements present in either set B or set C or both.
03

Illustrate A 鈭� (B 鈭� C).

Now, using the shaded region from Step 2, move onto finding A 鈭� (B 鈭� C). To do this, we will shade the region corresponding to the Union of set A and the already shaded region from the Union of sets B and C. This will include all the elements present in set A and the Union of sets B and C. Keep this diagram for comparison with the RHS Venn diagram.
04

Illustrate the Union of A and B.

In a new Venn diagram, again draw three circles representing sets A, B, and C. This time, shade the region corresponding to the Union of sets A and B (A 鈭� B), which includes all the elements present in either set A, set B, or both.
05

Illustrate (A 鈭� B) 鈭� C.

Using the shaded region from Step 4, move onto finding (A 鈭� B) 鈭� C. To do this, we will shade the region corresponding to the Union of the already shaded region from the Union of sets A and B, and set C. This will include all the elements present in the Union of sets A and B and set C.
06

Compare both Venn diagrams.

Now, compare both Venn diagrams showing the LHS (A 鈭� (B 鈭� C)) and RHS ((A 鈭� B) 鈭� C) of the given equation. Observe that both diagrams depict the same outcome: all the elements present in either of the sets or all three sets. Hence, the Venn diagrams illustrate that the given statement is true: $$ A \cup(B \cup C)=(A \cup B) \cup C $$

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