Venn Diagram - Algebra-1

1. Fundamental Concepts

Definition: A Venn diagram is a graphical representation used to show all possible logical relations between different sets.
Components: Circles or other shapes represent sets, and their intersections represent the common elements of those sets.
Purpose: To visualize relationships such as union, intersection, and complement of sets.

2. Key Concepts

Standard Regions:

a) $A \cup B$ : All areas covered by A or B (both circles combined).

b) $A \cap B$ : Only the overlapping region.

c) Complement (A'): Everything in U outside circle A.

3. Examples

Example 1 (Basic)

Problem: Consider two sets $A = \{1, 2, 3\}$ and $B = \{2, 4, 6\}$ . Draw a Venn diagram and find $A \cup B$ and $A \cap B$ .

Step-by-Step Solution:

Draw two overlapping circles representing sets A and B.
Place the numbers in the appropriate regions:
The union $A \cup B = \{1, 2, 3, 4, 6\}$ (all elements in A or B).
The intersection $A \cap B = \{2\}$ (where A and B overlap).

Example 2 (Intermediate)

Problem: Given four sets: U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = { $${x\in \mathbb{N} |x\text{ is even and }x\lt 10}$$ }, B = {1, 3, 5, 7} and C = {3, 6}. Draw a Venn diagram and find $$A \cap C$$ , $$B \cup C$$ , and the complement of A (A').

Step-by-Step Solution:

Identify the elements of set A:
Set A consists of natural numbers that are even and less than 10. Here are the elements: 2, 4, 6, 8.
Draw the Universal Set (U) as a rectangle and label circles:
Identify Regions:
$$A \cap C$$ = {6} (where A and C overlap).
$$B \cup C$$ = {1, 3, 5, 6, 7} (all elements in B or C).
A' = {1, 3, 5, 7, 9} (elements in U but outside A).

4. Problem-Solving Techniques

Label Clearly: Write elements in the correct regions.
Shade for Operations: Union ( $$\cup$$ ): Shade all involved circles. Intersection ( $$\cap$$ ): Shade only overlapping areas. Complement: Shade everything outside the set.