EQUALITY OF SETS

All girls in the class like John. All girls in the class like to eat pizza. The set of girls who like John is equal to the set of girls who like pizza.

Let A and B be sets. Consider that every element of set A is also an element of set B. Set A is equal to the set B if they both have the same members. The equality of sets A and B is denoted by A = B and with ( a A)( a A a B). This means that each element of the set A is also element of the set B. Here the symbol means ‘each’. The symbol means that if a A, then a B. We read it "a A implies a B" or " from a A outcomes a B".

Example: The set {a, b, c, d} is equal to the set {c, a, d, b}, i.e.

{a, b, c, d} = {c, a, d, b}.

The set {a, b, c} is different from the set {a, b}, i.e.

{a, b, c} {a, b}.

Let N be the set of all natural numbers. If S = {x | x N, x < 4} and A = {1, 2, 3},

Then S = A.

NULL SET

Let S be the set of Blondies who are clever. The set S is empty. (Remark: This is just a joke, as we know that IQ does not depend on the colour of the hair.)

Let N be the set of all natural numbers. Then the set {x | x N, x2 = 6} is an empty set because there is no natural number whose square is 6.

An empty set is a set with no elements. We denote it by the symbol and call it the null set.