# Difference between equals and equals

## Main difference - same vs. equivalent

Equal and equivalent are terms that are often used in mathematics. The main difference between equal and equal is that the term equal refers to things that are similar in all aspects , while the term equal refers to things that are similar in one particular aspect . Notice that in set theory, the words “equal” and “equal” have specific meanings, as we'll see below.

## What does it mean?

In general, two things are the same if they are similar in every way.

In 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 example, let's say $A=\left \{ 1,2,3,4 \right \}$ and $B=\left \{ 2,4,3,1 \right \}$ then,

the sentence $A$ is equal to the amount $B$ .

## What does equivalent mean?

Two things can be said to be equivalent if they are similar under a certain condition. If two entities are equivalent, therefore, depends largely on the condition that we use to describe their equivalence. For example, the numbers 2 and 7 are equivalent in the sense that they are both prime numbers. However, if we are interested in whether numbers are even, then 2 and 7 are not equivalent in that sense. We use the symbols $X\sim Y$ or $X\equiv Y$ to point out $X$ and $Y$ are equivalent.

Once a criterion is defined, things that are equivalent satisfy the equivalence relations :

1. Reflexivity : $X\sim X$
2. Symmetry : if $X\sim Y$ , then $Y\sim X$
3. Transitivity : If $X\sim Y$ and $Y\sim Z$ , then $X\sim Z$

In set theory, two sets are equivalent if they have the same number of elements. The elements themselves do not have to be the same either, only the number of elements has to be the same. For example, let's say $C=\left \{ 1,2,3,4 \right \}$ and $D=\left \{ 8,2,7,150 \right \}$ then,

The sentences $C$ and $D$ are equivalent. Symbols to express equality and equality

## Difference between equals and equals

### General meaning

When two things are the same , they are similar in all aspects.

When things are equivalent , they are similar in one 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. 