# Set Theory: Equal and Equivalent Sets

In these lessons, we will learn about equal sets and equivalent sets.

### Equal Sets

Two sets, P and Q, are equal sets if they have exactly the same members. Each element of P are in Q and each element of Q are in P. The order of elements in a set is not important.

Example:
List the elements of the following sets and show that P ≠ Q and Q = R
P = {x : x is a positive integer and 5x ≤ 15}
Q = {x : x is a positive integer and x 2 < 25}
R = {x : x is a positive integer and x ≤ 4}

Solution:
5x ≤ 15 ⇒ x ≤ 3 So, P = {1, 2, 3}
x 2 < 25 ⇒ x < 5 So, Q = {1, 2, 3, 4}
R = {1, 2, 3, 4}
Therefore, P ≠ Q and Q = R.

Equal sets, equivalent sets, one-to-one correspondence and cardinality
Two sets are equivalent if they have the same number of elements.
The elements do not need to be the same.
Equivalent sets have one-to-one correspondence to each other.
The cardinality of a set is the number of elements in the set.

What are Equivalent Sets?

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