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Last modified on July 10th, 2024

The intersection of two or more sets is a set that contains all the elements common to the original sets.

In set theory, the intersection of sets is denoted by the symbol â€˜âˆ©.â€™ The intersection of â€˜nâ€™ number of sets is expressed as Set 1 âˆ© Set 2 âˆ© Set 3 âˆ© â€¦ âˆ© Set n.

The formula for the intersection of two sets (A intersection B) is given by

**A âˆ© B = {x | x âˆˆ A and x âˆˆ B}**

This means x is an element of A âˆ© B if and only if x is an element of A and x is an element of B.

If A and B are two sets, then the intersection of A and B is written as A âˆ© B and read as â€˜A intersection B.â€™

If A = {3, 7} and B = {4, 7, 9} are two sets, then A âˆ© B = {7}, which is the only element common to sets A and B.

**Find the intersection of sets A and B when A = {3, 5, 7, 9, 11} and B = {4, 8, 12, 16}**

Solution:

Given, A = {3, 5, 7, 9, 11} and B = {4, 8, 12, 16}

Since there are no common elements between the two given sets

A âˆ© B = { } = É¸

Thus, the sets are disjoint.

If A, B, and C are three sets, then the formula to find the cardinality of the intersection of two sets is given by:

n(A âˆ© B) = n(A) + n(B) – n(A âˆª B)

Here,

n(A âˆª B) = Cardinality of set A âˆª B

n(A) = Cardinality of set A

n(B) = Cardinality of set B

n(A âˆ© B) = Cardinality of set A âˆ© B

If A, B, and C are three sets, their intersection is written as A âˆ© B âˆ© C. It is the set whose all elements are common to A, B, and C.

The formula for the intersection of three sets (A intersection B intersection C) is given by

**A âˆ© B âˆ© C = {x | x âˆˆ A and x âˆˆ B and x âˆˆ C}**

If A = {3, 7, 9}, B = {7, 9, 11, 13}, and C = {4, 7, 9} are three sets, then A âˆ© B âˆ© C is the element common to the sets A, B, and C.

Here, A âˆ© B âˆ© C = {7, 9}, which are the two elements common to sets A, B, and C

**Given: A = {6, 8, 9, 11, 12, 14, 15}, B = {8, 9, 11, 13, 14}, and C = {9, 11, 13, 15}, then find A âˆ© B âˆ© C**

Solution:

Given, A = {6, 8, 9, 11, 12, 14, 15}, B = {8, 9, 11, 13, 14}, and C = {9, 11, 13, 15}

Now,

A âˆ© B = {6, 8, 9, 11, 12, 14, 15} âˆ© {8, 9, 11, 13, 14} = {8, 9, 11, 14}

Now, (A âˆ© B) âˆ© C = {8, 9, 11, 14} âˆ© {9, 11, 13, 15} = {9, 11}

Thus, A âˆ© B âˆ© C = {9, 11}

In the Venn diagram, the shaded region illustrates the intersection of sets A and B. It covers all the elements common to both sets.

Similarly, a Venn diagram can be drawn for the intersection of three sets: A, B, and C.

The complement of the intersection of sets is a set containing all the elements in the universal set but absent in A âˆ© B. It is represented by (A âˆ© B)’ and is read as â€˜A intersection B complement.â€™

In the Venn diagrams below, the shaded regions mark the complement of the intersection of three sets: A, B, and C.

For example,

If A = {8, 16, 25}, B = {10, 12, 16}, and U = {6, 8, 10, 12, 16, 20, 25}, then A âˆ© B = {16}

Then, (A âˆ© B)’ = U – (A âˆ© B) = {6, 8, 10, 12, 16, 20, 25} – {16} = {6, 8, 10, 12, 20, 25}

If A and B are two sets, then according to the commutative law

**A âˆ© B = B âˆ© A**

**Verify the commutative law for the intersection of two sets, A and B, when A = {u, v, w, x} and B = {w, x, y, z}**

Solution:

Here, A = {u, v, w, x} and B = {w, x, y, z}

Now, A âˆ© B = {u, v, w, x} âˆ© {w, x, y, z} = {w, x}, andÂ

B âˆ© A = {w, x, y, z} âˆ© {u, v, w, x} = {w, x}

Thus, A âˆ© B = B âˆ© A, verified.

If A, B, and C are three sets, then according to the associative law

**(A âˆ© B) âˆ© C = A âˆ© (B âˆ© C)**

**Verify the associative law for the intersection of three sets, A, B, and C. Given A = {2, 4, 6, 8}, B = {4, 6, 12}, and C = {2, 4, 8}**

Solution:

Given, A = {2, 4, 6, 8}, B = {4, 6, 12}, and C = {2, 4, 8}

Now, A âˆ© B = {2, 4, 6, 8} âˆ© {4, 6, 12} = {4, 6}

B âˆ© C = {4, 6, 12} âˆ© {2, 4, 8} = {4}

(A âˆ© B) âˆ© C = {4, 6} âˆ© {2, 4, 8} = {4} and A âˆ© (B âˆ© C) = {2, 4, 6, 8} âˆ© {4}

Thus, the associative law is verified.

If A, B, and C are three sets, then according to the distributive law for the intersection of sets

**A âˆ© (B âˆª C) = (A âˆ© B) âˆª (A âˆ© C)**

The intersection of any set with an empty set produces the empty set.

**A âˆ© É¸ = É¸ **

If A = {2,10, 25} and É¸ is an empty set, then A âˆ© É¸ = {2, 10, 25} âˆ© { } = { } = É¸

The intersection of any set with the universal set produces the original set.

**A âˆ© U = U**

If A = {2,10, 25} and U = {1, 2, 5, 10, 15, 20, 25}, then A âˆ© U = {2,10, 25} âˆ© {1, 2, 5, 10, 15, 20, 25} = {1, 2, 5, 10, 15, 20, 25} = U

It states that the intersection of a set with itself results in the same original set.

**A âˆ© A = A**

**Given A = {19, 38, 57}. Verify the idempotent law for the intersection.**

Solution:

Given, A = {19, 38, 57}

A âˆ© A = {19, 38, 57} âˆ© {19, 38, 57} = {19, 38, 57} = A

Thus, the idempotent law for intersection is verified.

Union of Sets | Intersection of Sets |
---|---|

The union of sets is a new set containing all the given set elements. | The intersection of sets is a new set that contains all common elements of the given sets. |

Denoted by the symbol â€˜âˆª.â€™ | Denoted by the symbol â€˜âˆ©.â€™ |

Written as A âˆª B and read as â€˜A union B.â€™ | Written as A âˆ© B and read as â€˜A intersection B.â€™ |

In set builder notation, it can be expressed asA âˆª B = {x | x âˆˆ A or x âˆˆ B} | In set builder notation, it can be expressed asA âˆ© B = {x | x âˆˆ A and x âˆˆ B} |

Example:If A = {3, 13} and B = {3, 7, 17}, then A âˆª B = {3, 7, 13, 17} | Example:If A = {3, 13} and B = {3, 7, 17}, then A âˆ© B = {3} |

Last modified on July 10th, 2024