Definition Of Equality Of Sets

Let a and b be sets.

Definition of equality of sets. Mathematically it can be written as a b and b a. First we check every in the left. Section 7 2 equality of functions two functions are equal if they have the same domain and codomain and their values are the same for all elements of the domain. Thus for example 1 2 3 3 2 1 that is the order of elements does not matter and 1 2 3 3 2 1 1 that is duplications do not make any difference for sets.

A is equal to b written a b if a b and b a. The identity relation is an equivalence relation. We can prove this using the definition. It is very important to note that to prove that two sets are equal we must show that both sets are subsets of each other.

There s no way sets a and b can have the same objects but set b and a can have different objects. 1 2 3 4 and 3 4 2 1 are equal. It must work both ways each element of the 1st set must be in the 2nd set and each element of the 2nd set must be in the 1st set. Viewed as a relation equality is the archetype of the more general concept of an equivalence relation on a set.

Let a and b be sets and f a to b and g a to b be functions. Equality sets two sets and are equal if for all if and only if. Equality sets more. This definition tells us that sets ignore duplicates and order.

In this case we write it as a b. Since a set is just a collection of objects 1 2 3 and 1 1 2 2 3 3 are both sets but by the definition of equality they are the same set that is 1 2 3 1 1 2 2 3 3. Two sets are equal equal when they contain only the same things. Equality of sets is defined as set a is said to be equal to set b if both sets have the same elements or members of the sets i e.

Likewise if a and c both have the same objects as b they can t have different objects. We say that f and g are equal and write f g if f a g a for all a in a text if f and g are not equal we write f ne g text. Those binary relations that are reflexive symmetric and transitive. Definition equality of sets.

These three definitions are equivalent proof left as an exercise so you can use them interchangeably. Sets that have precisely the same elements. Two sets are equal if and only if they have the same elements. The sets and are said to be equal if and denoted by.

They don t have to be in the same order.