Operations on sets are somewhat similar to operations of addition, multiplication and subtraction of numbers.

Let A and B be two sets.

The set of elements that are in either A or B or both is called the union of the sets A and B and is denoted by A ? B, i.e.,

A ? B = {x| x ? A or x ? B}

The set of all elements of the sets A and B is called the union of the sets A and B and is denoted by

 A ? B.

Union is the act of combining two sets together into a single set.

Example

A = {1, 3, 5, 8}, B = {1, 3, 7}. Then A ? B = {1, 3, 5, 7, 8}.

If an element appears in both sets then we only list it once in the new set.

Example

A = {x| x is a city in Turkey with population greater than 1 million}, B = {x| x is a city in Turkey with population less than 500 000}. Then A ? B ={x| x is a city in Turkey with population greater than 1 million or less than 500 000}. The set of elements A which are not in B is called the difference between A and B and is denoted by A \ B.

A \ B = {x| x ? A and x ? B}

Example

A = {3, 5, 8, 10}, B = {4, 5, 9}. Then A \ B = {3, 8, 10}.

Example

A = {0, 1, 2, 3, 4, …}, B = {1, 3, 5, 7, …}. Then A \ B = {0, 2, 4, 6, ...}.

Usually the sets that we deal with are subsets of some ambient set. Such a set is called a universal set and is denoted by U. In other words, U is the universal set if all the sets under examination are subsets of U. The difference U \ A is called the complement of A and is denoted by A^c . That is,

A^c = U \ A = {x| x ? U and x ? A}

Example

U = {1, 2, …, 10}, A = {9, 10}. Then U \ A = {1, 2, …, 8}.

The intersection of two sets A and B, written A ? B is the set consisting of the elements of both A and B. Thus, A ? B = {x| x ? A and x ? B}

Example A = {1, 2, 3, 5, 8}, B = {2, 3, 10, 11}. Then

A ? B = {2, 3}. x ? A ? B if and only if x ? A and x ? B.

Example

A = {x| x is a city in Turkey with population less than 1 million},

B = {x| x is a city in Turkey with population greater than 500 000}.

Then A ? B ={x| x is a city in Turkey with population between 500 000 and 1 million}.

As also understood from the information given,

If A = {1, 2, 3, 5, 8} and B = {2, 3, 10, 11}, the expressions in the options;

I “A ? B = {2, 3}”

III “A \ B = {1, 5, 8}”

IV “B \ A = {10, 11}” are correct, so the correct answer is D.

If an element appears in both sets then we only list it once in the new set, so the expression in the option II “A ? B = {1, 2, 2, 3, 3, 5, 8, 10, 11}” is not correct. It is written as;

A ? B = {1, 2, 3, 5, 8, 10, 11}.