Understanding Equivalent Concepts in Maths

If your mother gives two chapatis to your elder brother, and one to you, is it unfair? NO, right? Because he is older than you, he needs more food. After having food, both of your stomachs filled completely. This is what we call ‘Equivalent’. 

An equivalent is basically the numbers or quantities which are the same or logically the same. ‘Logically same’ means that numbers might be different but eventually the same part of a whole. To know more about the term ‘Equivalent’ in Mathematics, read the complete article.

Equivalent- Meaning in Mathematics

The term "equivalent" in math refers to two meanings, numbers, or quantities that are the same. The equivalence of two such quantities shall be denoted by a bar over an equivalent symbol or Equivalent Sign. It also means a logical equivalence between two values or a set of quantities. Equivalence is similar but more universal than equality. If two sets of equations have the same solutions, we might consider them similar, but they are not identical.

How to Define Equivalent in Mathematics?

In mathematics, Equivalent meanings are used in two different ways. First, within the framework of a particular mathematical theory (An Equivalent Example would be Euclidean geometry), a notion (e.g. ellipse or minimal surface) may have more than one meaning. In the context of a given mathematical form, these concepts are identical (Euclidean space, in this case). Second, there could be a mathematical framework. In the prior example, the equivalence of two definitions implies that a mathematical entity follows one definition if and only if it meets the other definition. In the above case, the sense of equivalence is more complex since the structure is more abstract than the entity. Several different artifacts can follow the same structure.

Representation of Equivalent Symbol

If A ⇒ B and B ⇒ A (i.e., A ⇒ B & B ⇒ A, where ⇒ denotes implies), then A and B are said to be equal, a relationship that is symbolically written as A = B in this job. 

• Equivalent Expressions: An algebraic expression is an expression composed of variables, coefficients, constants, and mathematical functions, such as addition, subtraction, multiplication, and division. Generally, if two objects are the same, they're considered identical. Similarly, in mathematics, equivalent expressions are expressions that are the same, even though the expression looks different. Look at 3 × 3 + 1 and 5 × 2 expressions. They're both equivalents to 10. That is, they are equivalent expressions.

How to find that two mathematical expressions are equivalent expressions or not? 

Ans. In general, to prove that two mathematical expressions are equivalent we keep them as equal. Later, we evaluate the left-hand side expression and right-hand side expression, and if both sides are equal then the given expressions are equivalent expressions. (To solve the basic mathematical functions we use the BODMAS–Bracket OF Division Multiple Addition Subtraction rule.)

For Example: Check whether 3 × 9 + 5 × 2 is equivalent to 7 × 3 + 4 × 4 or not.

Solution: To check the equivalency of both the equations, we put both of them equal to each other.

3 × 9 + 5 × 2 = 7 × 3 + 4 × 4

27 + 10 = 21 + 16

37 = 37

Therefore, L.H.S. = R.H.S.

So, the given expressions are equivalent to each other. 

Remember: We also need to give attention to the units obtained on both sides. 

• Equivalent Fractions:  Equivalent fractions say that when all results are the same fraction after simplification, two or more than two fractions are said to be equal. Let's assume, after simplifying these fractions, a/b and c/d are two fractions, both result in identical fractions, say e/f, then they are equal to each other.

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Equivalent fractions are basically the same part of a whole. For example: \[\frac{1}{2}, \frac{2}{4}, \frac{3}{6}...,\frac{36}{72}...,\]are the example of equivalent fractions. If we solve and simplify each term, it results \[\frac{1}{2}\] .

By multiplying the numerator and denominator with the same number, you can find equivalent fractions of any given fraction. And, by dividing the numerator and denominator with the same number, you can also find the equivalent fraction.

• Equal and Equivalent Sets: Equivalent sets should have the same cardinality, to be equivalent. This suggests that one-to-one correspondence can be made between the elements of both sets. Here, one-to-one correspondence implies that an element remains in set B with each element in set A before the sets are exhausted. What is an Equal set? In the case of equal sets only if each element of set A is also an element of set B will the two sets A and B be equal. They are often said to be equivalent if two sets are each other's sub-sets. This shall be expressed by:

A = B 

A ⊂ B and B ⊂ A ⇔ A = B

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The first image represents an equal set and the second image represents an equivalent set.

In simple words, if A={2, 4, 6, 8, 10}, B={@, #, %, &, !}, C={1, 2, 3, 4, 5, 6}, D={2, 4, 6, 8, 10}, then set A, set B and set D are said to be equivalent sets, because all three sets have 5 elements. Therefore equivalent sets are those sets that have an equal number of elements, irrespective of what they are. 

Remember: All equal sets are equivalent sets. Also, all null sets (sets that have no elements) are equivalent sets.

• Equivalent Ratios: Two ratios that express the same relationship between numbers are identical ratios or identical ratios. We multiply or divide the terms of the given ratio by the same non-zero number to get a ratio equal to a specified ratio. Equivalent ratios have varying numbers, but the same relationship is reflected.

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How to find that two ratios are equivalent or not? 

Two or more than two ratios can be compared with each other. In order to find whether they are equivalent or not, first, we convert them into like fractions. After converting, denominators of all the fractions become equal and if the numerators of all the fractions also become equal, the fractions are said to be equivalent fractions. 

Also, if the two fractions a/b and c/d become equivalent, then the four quantities a, b, c, and d are said to be in proportion. It can be written as a:b::c:d.

Solved Examples

1. Are 4/5 and 16/20 equal ratios?

Solution: If we simplify 16/20 we get 4/5, as 16 and 20 both when divided by 4 gives 4 and 5 respectively.

Therefore the above ratios are equivalent.

2. Identify a fraction that is equivalent to 7/9.

Solution: In Equivalent Calculations Maths, any equivalent fraction can be obtained by multiplying and dividing the same number into the fraction. Therefore let us multiply 2 to both numerator and denominator.

On multiplying 2 to both numerator and denominator, we get,

(7 × 2)/(9 × 2) = 14/18.

3. Check whether 7 × 6 + 66 ÷ 11 - 5 × 2 is equivalent to 7 × 3 + 24 ÷ 2 + 9 × 3 or not.

Solution: To check equivalency of both the equations, we put both of them equal to each other.

 7 × 6 + 66 ÷ 11 - 5 × 2 = 7 × 3 + 24 ÷ 2 + 9 × 3

42 + 6 - 10  = 21 + 12 + 27

38 ≠ 60

Therefore, L.H.S. ≠ R.H.S.

So, the given expressions are not equivalent to each other.