## Main Difference – Equal vs. Equivalent

**Equal** and **equivalent** are terms that are used frequently in mathematics. The **main difference** between equal and equivalent is that the term **equal refers to things that are similar in all aspects**, whereas the term **equivalent refers to things that are similar in a particular aspect**. Note that in the **set theory**, the words “equal” and “equivalent” have specific meanings, as we will see below.

## What Does Equal Mean

In general, two things are equal if they are similar in all respects.

In the case of set theory, two sets are equal if they both contain the same elements. The *order* in which they are listed in a set does not matter. For instance, suppose

and

then,

the set is equal to the set .

## What Does Equivalent Mean

Two things can be said to be equivalent if they are similar under a particular condition. Therefore, whether two entities are *equivalent* largely *depends* on the condition we use to describe their equivalence. For instance, numbers 2 and 7 are equivalent in the sense that they are both prime numbers. However, if the condition we are interested in is finding out whether numbers are even, then in this sense 2 and 7 are *not* equivalent. We use the symbols or to indicate that and are equivalent.

Once a criterion has been defined, things that are equivalent satisfy the **equivalence relations**:

**Reflexivity**:**Symmetry**: If , then**Transitivity**: If and , then

In set theory, two sets are *equivalent* if they have the same number of elements. The elements themselves do not need to be the same, either, *only* the number of elements needs to be the same. For instance, suppose

and

then,

the sets and are equivalent.

## Difference Between Equal and Equivalent

### General Meaning

When two things are **equal**, they are similar in all aspects.

When things are **equivalent**, they are similar in a particular aspect.

### In Set Theory

When two sets are **equal**, they contain the same elements.

When two sets are **equivalent**, they contain the same *number of* elements.