Equal and Equivalent Sets

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Define Equal Sets

To understand Equal Set meaning, Equal Set is defined as two sets having the same elements. Two sets A and B can be equal only on the condition that each element of set A is also the element of set B. Also, if two sets happen to be the subsets of each other, then they are stated to be equal sets.

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Equal Sets

An equal set can be represented by:

P = Q

P ⊂ Q and Q ⊂ P ⟺ P equals to Q

It is to be noted that if the condition discussed above is not met, then the set is stated to be unequal.

Unequal sets are represented by

P ≠ Q

Define Equivalent Sets

Equivalent sets meaning in Mathematics holds two definitions.

Equivalent Sets Definition 1 - Let's say that two sets A and B have the same cardinality, then, there exists an objective function from set A to B.

Equivalent Sets Definition 2 - Let's say that two sets A and B are stated to be equivalent only if they have the same cardinality, that is, n(A) = n(B).

Thus, to remain or be equivalent, the sets should possess the same cardinality. This condition means that there should be one to one correspondence between the elements belonging to both the sets. In this context, the one to one condition implies that for each element on the set A, there exists an element in the set B, till both the set A and set B gets exhausted.

Therefore, in general, it can be stated that the two sets remain equivalent to each other if only the number of elements in both the sets remain equal. The sets don't need to hold the same elements, or they stay to be a subset of each other.

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Equivalent Sets

Equal And Equivalent Sets Examples

Equal Set Example

Let’s understand equal sets with an example,

If M= {1, 3, 9, 5, −7} and N = {5, −7, 3, 1, 9,}, then it can be stated that M = N. It is to be noted that no matter how many times an element is repeated in a particular set, the element is counted only once. Also, it is to be pointed out that the order does not matter for the elements for a specific set. Therefore, in terms of cardinal number, equal sets can be stated that: 

If P = Q, then n(P) = n(Q) and for any x ∈ P, x ∈ Q too.

Equivalent Set Example

If S = {x: x, where x is stated to be a positive integer} and T = {d : d, where x is said to be a natural number}, then S is stated to be equivalent to T.

Thus, it can be stated that an equivalent set is simply a set with an equal number of elements. However, the sets don't need to have the same elements but must comprise the same number of elements. 

Let’s Understand Equivalent Sets With Examples

  1. If A = {1,−7,200011000,55} and B = {1,2,3,4}, then A is equivalent to B.

  2. If Set G: {Sweater, Mittens, Scarf, Jacket} and Set H: {Apples, Bananas, Peaches, Grapes}, it can be noted that both Set G and Set H comprise word elements in different categories and have the same number of elements i.e. four. 

Important Points to Remember on Equivalent Sets

  • All the null sets are said to be equivalent to each other.

  • Not all the infinite sets remain equivalent to each other. For example, the equivalent set of all the real numbers and the equivalent set of the integers. 

  • If P and Q are stated to be two sets such that P is equal to Q, that is, (P = Q). This example means that two equal sets will always remain to be equivalent, but the converse of the equivalent set may or may not remain true.

  • An equal set can be an equivalent set, but it is not necessary for an equivalent set to be an equal set.

FAQ (Frequently Asked Questions)

1. Define equal set and equivalent set with example.

The equal set definition is that when two sets have the same elements. However, it does not matter which order the elements are arranged. The only thing that matters in an equal set is that the same elements are present in each set.

For example: {2, 4, 6, 8} and {8, 6, 4, 2} or {Blue, Pink, Green, Orange} and {Orange, Green, Pink, Blue}

The equivalent set definition states that in a simple set, there is an equal number of elements. Equivalent sets do not have to hold the same number but the same number of elements. 

For example Set P: {F, G, H, I, J} and Set Q: {June, July, August, September, October}

Sets P and Q comprise completely different elements (Set P contains letters, and Set Q includes months of the year). However, they have the same amount of elements, which is five. This feature makes them equivalent.

2. What are the primary differences between equal and equivalent sets?

Equal set definition math states that when two sets have the same and equal elements, they are called Equal Sets. The arrangement or the order of the elements does not matter, only the same elements in each set matter. A suitable example of an equal set is {January, March, May, November} and {May, March, January, November}.

Equivalent set meaning states that two sets comprise an equal number of elements. It is not necessary to hold the same elements but include the same number of elements. A suitable example of equivalent sets is Set A: {M, N, O, P, Q, R} and Set B: {Red, Blue, Green, Yellow, Pink, Purple}

Sets A and B comprise completely different elements (Set A includes letters, and Set B includes colours). However, they have the same amount of elements i.e. six. 

An equal set is an equivalent set, but an equivalent set necessarily cannot be an equal set.