Operations on Sets Union Intersection: Two sets are disjoint if their intersection is the null set. Difference A - B. Complement of a set A.

Union If A and B are sets, their union can be defined as the set that consists of all elements of A or B. It is denoted by A U B. A U B = {x | x Є A or x Є B} A B A U B Venn Diagram

Union: Examples Example 1: A = {1, 2, 5, 6,7} and B = {3,4, 5, 6, 8, 9} A U B = {1, 2, 3,4, 5, 6, 7, 8, 9} Example 2: A = {1, 2, 3, 4} A U A = {1, 2, 3, 4} Example 3: A = {1, 2, 3, 4} A U Ø = {1, 2, 3, 4} or A U { } = {1, 2, 3, 4}

Intersection If A and B are sets, their intersection can be defined as the set that consists of all elements that belong to both A and B. It is denoted by A ∩ B. A ∩ B = {x | x Є A and x Є B} A ∩ B Venn Diagram A B

Intersection: Examples Example 1: A = {1, 2, 5, 6,7} and B = {3,4, 5, 6, 8, 9} A ∩ B = {5, 6} Example 2: A = {1, 2, 3, 4} A ∩ A = {1, 2, 3, 4} Example 3: A = {1, 2, 3, 4} A ∩ Ø = Ø or A ∩ { } = { }

Disjoint Sets Two sets that have no common elements are called disjoint sets. Example: A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 10, 12} A ∩ B = { } AB Venn Diagram A ∩ B = { }

Complement If A and B are two sets, the complement of B with respect to A is defined as the set of all elements that belong to A but not to B. It is denoted by A – B. A – B ={x | x Є A and x Є B B – A ={x | x Є B and x Є A A B B - A A B A - B

Complement If U is a universal set containing A, then U – A is called the complement of A. It is denoted by Ā. Ā = {x | x Є A} Example: A = {x | x is an integer and x ≤ 8} and U = Z (Z ->set of all integers). Ā = {x | x is an integer and x > 8}

Properties of Complement Ā = A A U Ā = U A ∩ Ā = Ø Ø = U U = { } A U B = Ā ∩ B A ∩ B = Ā U B

Symmetric difference If A and B are two sets, their symmetric difference can be defined as the set of all elements that belong to A or to B, but not to both A and B. It is denoted by A O B. A O B ={x | (x Є A and x Є B) or (x Є B and x Є A)} A O B = (A – B) U (B – A) A B B - A

Symmetric Difference: Example Example1: A = {1, 2, 3, 4} and B = {3, 4, 5, 6, 7} A O B = {1, 2, 5, 6} Example 2: A = {a, c, f} and B = {f, h, k} A O B = {a, c, h, k} Example 3: A = {2, 4, 5, 9} and C ={x| x is a positive integer and x 2 ≤ 16} A O B = {5, 9, 3, 1}

1.2 Exercises Identify the following as true or false.  y Є A ∩ B x Є B U C  w Є B ∩ C u Є C w y v x z u U A B C

The Addition Principle |A U B U C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C| A B C |A ∩ B| |B ∩ C| |A ∩ C| |A ∩ B ∩ C| Inclusion-exclusion principle:

1.2 Exercises A survey of 500 television watchers produced the following information: 285 watch football games, 195 watch hockey games, 115 watch basketball games, 45 watch football and basketball games, 70 watch football and hockey games, 50 watch hockey and basketball games, and 50 do not watch any of the three kinds of games. a) How many people in the survey watch all three kinds of games? b) How many people watch exactly one of the sports?

1.2 Exercises |F U H U B| = |F| + |H| + |B| - |F ∩ H| - |H ∩ B| - |F ∩ B| + |F ∩ H ∩ B| F H B |F ∩ H| |H ∩ B| |F ∩ B| |F ∩ H ∩ B|

1.2 Exercises Given: |F| = 285, |H| = 195, |B| = 115, |F ∩ H| = 70, |H ∩ B| = 50, |F ∩ B| = 45, |F U H U B| = 500 – 50 = 450 |F U H U B| = |F| + |H| + |B| - |F ∩ H| - |H ∩ B| - |F ∩ B| + |F ∩ H ∩ B| F H B |F ∩ H| |H ∩ B| |F ∩ B| |F ∩ H ∩ B|

1.2 Exercises Given: |F| = 285, |H| = 195, |B| = 115, |F ∩ H| = 70, |H ∩ B| = 50, |F ∩ B| = 45, |F U H U B| = 500 – 50 = 450 |F U H U B| = |F| + |H| + |B| - |F ∩ H| - |H ∩ B| - |F ∩ B| + |F ∩ H ∩ B| or, 450 = – 50 – 45 + |F ∩ H ∩ B| or, |F ∩ H ∩ B| = 20 F H B |F ∩ H|= 70 |H ∩ B|= 50 |F ∩ B| = 45 |F ∩ H ∩ B|= 20 Ans. for 24(a)

1.2 Exercises Given: |F| = 285, |H| = 195, |B| = 115, |F ∩ H| = 70, |H ∩ B| = 50, |F ∩ B| = 45, |F U H U B| = 500 – 50 = 450 From 24(a): |F ∩ H ∩ B| = 20 No. of people watching only Football = 285 – 70 – = 190 No. of people watching only Hockey= 195 – 70 – = 95 No. of people watching only Basketball = 115 – 45 – = 40. Total no. of people watching exactly one of the sports = = 325 F H B |F ∩ H|= 70 |H ∩ B| = 50 |F ∩ B| = 45 |F ∩ H ∩ B| Ans. for 24(b)