In the set theory of mathematics, equal sets are sets with the same number of elements. Note that the elements present in equal sets are identical. Inset theory, the concept revolves around set equality. Before getting into the details of set theory, we find it is essential to ease the concept of Set. Set refers to the collection of well-defined elements.
It could be letters, numbers, shapes, and people. Elements in a state get denoted by braces {} and capital letters. The set theory revolves around the operational principles of sets.
The course will start with the definitions of equal sets and explore the properties of like sets. One needs to know the differences between homogeneous and heterogeneous sets. The concept will become clear with examples of different forms of sets.
What are equal sets?
Equal sets are sets with equal elements and the same cardinality. In simple terms, sets are equal only when they have the same number of identical elements. Let us now learn equal set with an example.
P is a set with elements {1, 2, 3, 4, 5}, O is an equal set of P when it has elements {1, 2, 3, 4, 5}. As O and P have the same elements and cardinality, they are equal sets.
If two sets don’t have the same number of elements, they form unequal sets.
What are equivalent sets?
If two sets have an equal number of elements, they are equivalent sets. If O = {1, 2, 3, 4, 5} and P ={ 2, 3, 4, 5, 6}, O and P are equivalent sets but not equal. If O = {1, 2, 3, 4, 5} and U = { 2, 3, 4, 5}, O and P don’t have the same cardinality. It means that P and O are unequal sets. Since O and P don’t have the same elements, they are unequal sets.
How can one define equal sets?
Note that if the number of elements of two or more sets is equal and the elements are equal, sets are equal. Symbol or sign used to denote equal set is =. If D and E are equal sets, then one can write it as D = E.
If D = {a, b, c, d} and E = {a, c, b, d}, D and E are equal sets. The orders in which elements appear don’t matter in sets. Sets are equal when they have the same number of elements, and elements in the sets are equal.
What are Venn diagrams?
Venn diagrams get used to represent equal sets. A Venn diagram gets used to represent equal sets. M and N are equal sets when they have the same number of elements. Here is a Venn diagram to represent two sets M and N with the elements {12, 14, 28}
M = {12, 14, 28} = N
M N
We can represent the two equal sets M and N with the same elements 12, 14, and 28.
Some of the properties exhibited by equal sets
By now, you have got a fair idea of equal sets. It’s time to discuss some of the properties exhibited by equal sets that help clear doubts and make it easy to understand the concept. Let’s identify and understand the properties of equal sets.
- The order of means in which elements appear in a set does not impact its equality.
- Equal sets have the same cardinality. It means that equal sides have the same number of elements.
- · If two sets are subsets of each other, two sets are equal to each other. If M ⊆ N and N ⊆ M, it means that N = M.
- · Two sets are equal when they have an equal number of elements.
- All equal sets are equivalent sets. But all equivalent sets and not equal sets. Now that you have an idea of equal and equivalent sets, it is time to know their differences.
The table highlights the difference between equivalent and equal sets. It is how an equivalent and equal set differ from one another.
- If all the elements are equal in two or more sets, they are equal.
- If the number of elements is equal in one or more sets, they are equivalent.
- Equal sets have the same cardinality.
- Equivalent sets have the same cardinality.
- Equal sets have the same number of elements.
- Equivalent sets have the same number of elements.
- The symbol used in indicating equal sets is =.
- The symbol used in indicating equivalent sets is ≡ or ~.
- All equal sets are equivalent sets.
- All equivalent sets are not equal.
- The number and value should be the same in an equal set.
- While numbers should be the same, the equivalent set may not have the same value.
Things that you should take into account when learning about equal sets
- Sets with the same elements are equal.
- If two sets are subsets of each other, they are equal.
- Sets with the same elements are equal.
Show that C ={x: x is prime so that 1 < x < 10} B ={ 3, 7, 5, 2}
Solution: C ={x: x is prime so that 1 < x < 10} = { 3, 7, 5, 2} The number of elements in C is 4, and the sets are equal. Hence, C = B.
Answer: C ={x: x is prime so that 1 < x < 10} = { 3, 7, 5, 2} = B.
Frequently asked questions on equal sets.
How can one define equal sets?
Equal sets are sets with the same elements, and they have all equal elements. Equal sets are sets with the same cardinality and equal elements.
What are the differences between equivalent and equal sets?
If the two sets have the same elements, they are equal. Equivalent sets have the same number of elements, but the elements may not be the same.
How can one prove that two sets are equal?
To prove that two sets are equal, one must prove that they are subsets of one another. There is another way to prove that two sets are equal: you need to examine the equality and cardinality of the elements.
Are all equivalent sets equal?
All equivalent sets are not equal. Two equivalent sets are equal only when they have the same element.
Conclusion
We hope this helps you understand the equal sets and equivalent sets perfectly.
