Compute A^c – B with Solved Example and Venn Diagram | Query Point Official

Given that `U = {xIx \in W, \le x \le 20}`
`A = {x \in P, x \le 20}`
`B = {x \in N, x \le 15}`
Where P is the set of prime numbers and N is the set of Natural Numbers
Compute `A^c – B` and draw a Venn diagram. >

In set theory, one of the most important operations is the intersection of sets. The intersection of two sets contains all elements that belong to both sets. It is denoted by the symbol `\(A \cap B\)`. A Venn diagram visually shows the relationship between sets and highlights the intersection as the overlapping region between the circles representing each set.

Definition: Intersection of Sets

The intersection of two sets (A) and (B) is the set of all elements that are common to both sets. It is written as:

`\(A \cap B = \{x : x \in A \text{ and } x \in B\}\)`

In a Venn diagram, the intersection is the shaded area where the circles for sets A and B overlap.

Example Problem

Suppose:

A = {1, 2, 3, 4, 5}

B = {3, 5, 7, 9}

Find the intersection:

Compare the elements of both sets to see which elements appear in **both** A and B.

✔ Elements common to A and B are: 3 and 5.

Therefore, `\(A \cap B = \{3, 5\}`

Venn Diagram Representation

A Venn diagram uses overlapping circles to show how sets relate to each other. The overlapping area between two circles represents the intersection of the sets.

In this diagram:

• The left circle represents set A.
• The right circle represents set B.
• The overlapping middle part shows all elements that belong to A ∩ B.

When Is the Intersection Empty?

If two sets have no common elements, then:

`\(A \cap B = \varnothing\)`

Such sets are called disjoint sets.

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Frequently Asked Questions (FAQs)

Q1: What does `\(A \cap B\)` mean?

`\(A \cap B\)` means the intersection of sets A and B — the elements that are in both sets.

Q2: How do you draw a Venn diagram for two sets?

Draw two overlapping circles inside a rectangle (the universal set). Label each circle with the set name. The overlapping portion represents shared elements (intersection), and the outer parts show elements unique to each set.

Q3: Is the intersection always the same as the union?

No. The intersection contains only elements common to both sets, while the union contains elements that are in either set or both.

Q4: What symbol represents the intersection?

The symbol for intersection is ∩, read as “cap,” representing the overlapping or common area between sets.

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Explore more about DISCRETE MATHEMATICS in Mathematics Notes & MCQs.