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Equal and Equivalent sets are two types of sets based on the elements present in them. This article provides worksheet on equal and equivalent set to practice question that need to be solved before appearing in exams. This article also has important concept and solved examples that students should go through to get an idea how to solve worksheet on equal and equivalent sets.
Important Concepts on Equal and Equivalent Sets
The important concepts needed to solve questions on equal and equivalent sets are mentioned below:
Equal Sets
Two sets are said to be equal sets if they contain the same elements, regardless of the order in which these elements occur. If there are two sets, set A and set B, they are said to be Equal if every element a ∈ A is also present in B, and for every element b ∈ B is also in set B. It is represented as A = B.
For example: let A = {1, 5, 7} and B = {7, 1, 5}. In this case, A = B because both sets contain the same elements, even though the order is not the same.
Properties of Equal sets
- Reflexive property: It means every set is Equal to itself
- Symmetry: If two sets, A and B, are Equal, then A = B and B = A.
- Transitivity: If there are three sets A, B, and C, such that set A is equal to B and B is equal to C, then A is also equal to C.
- Two Equal sets are always subsets of one another.
- The elements of both sets need to be the same.
Equivalent sets
The concept of Equivalent sets is related to the cardinality of sets, which refers to the number of elements in a set. Two sets are set to be Equivalent if they have the same number of elements, irrespective of what those elements are. Hence, two sets, set A and set B are said to be Equivalent If the cardinality of A Is equal to the cardinality of B. It is represented as A ~ B.
For example: Let A = {2, 5, 7} and B = {a, b, c}. In this, A and B are equivalent sets because both the sets have three elements.
Important related formulas and concepts
- Representation of Equal sets = ∀ x { x ∊ A ⇔ x ∊ B}
- Representation of Equivalent = A ∽ B
- Cardinality: It is the number of elements present in a set. It is denoted by n(A).
- For example, for set A = {a, e, i, o, u}, n(A)= 5.
Properties of Equivalent sets
- Reflexive property: It means that every set is Equivalent to itself
- Transitivity : If there are three sets A, B, and C, and set A is Equivalent to set B and Set B is Equivalent to set C. Then, set A is also Equivalent to set C.
- Two Equal sets are said to be Equivalent if they have the same cardinal number.
- Two Equivalent sets may or may not be equal.
Worksheet on Equal and Equivalent Sets
Q1: Are the sets k = {x : x2 = 4} and l = {2, -2} Equal ?
Q2: C= {natural number} and D = {whole numbers} Are they Equivalent?
Q3: Is the Following set Equal or Equivalent: A= {Colours of the rainbow}, B = {Red orange, yellow, green, blue indigo violet}?
Q4: Are the sets O= {x : x Is a positive divisor of 6} and P = {1, 2, 3, 6} Equal ?
Q5: Are the sets U = {x : x = y2, y ∊ {1,3, 5} } and V = {1, 9, 25} Equal ?
Q6: Are the sets {H, E, L, O} and B= {3,7,8,9,10} Equivalent?
Q7: {pumpkin, papaya, peas, pigeon} and {2,3,5,7,9} Equivalent?
Q8: State true or false: All Equal sets are Equivalent.
Q9: State the cardinal number of set: {x : x<10, x ∊ N }
Q10: Are the sets M = {x : x is a prime number less than 20} and N = {2, 3, 5, 7, 11, 13, 17, 19} Equivalent?
Solved Examples on Equal and Equivalent Sets
Q1: State if the following are Equal sets or not.
P = {2, 4, 6, 8, 10}
Q = {x : x = 2y, y ∊ N, y < 6 }
Solution:
Both sets should have the same elements so that sets P and Q are Equal.
Let’s dissect set Q
So y is a natural number, and y < 6, so the natural numbers that satisfy it are 1, 2, 3, 4, 5.
Therefore Q = {2, 4, 6, 8, 10}
Since P and Q contain exactly the same elements, So, they are Equal sets.
Q2: Classify the following sets as Equal and Equivalent sets.
P = {letters in the word SINCE}, Q= {SUN, MON, TUES }, R= {VERTICES OF A TRIANGLE ABC}, S = {LETTERS IN CAB}, T = {WED, THU, FRI}, U= {letter in the word SCIENCE}
Solution:
For two sets to be Equal, both the sets should have the same elements, and for them to be Equivalent, they should have the same cardinal number.
Equal sets
Sets P and U
As they contain the same elements.
Equivalent sets
Sets Q and T
Sets R and S
As they contain the same number of elements
Q3: Are the sets: A = {x : x2 – 4x + 6 = 0} and B = {x : 2x2 – 8x + 12 = 0} Equal ?
Solution:
Solving both the equations we get A = {1, 3} and B = A = {1, 3}. As both the sets have the same elements, they are Equal.
Q4: Check if the following sets are Equal or not.
X = {5, 10, 15} and Y = {10, 5, 15}.
P = {apple, orange, banana} and Q = {banana, orange, apple}.
M = {2, 4, 6} and N = {3, 6, 9}.
Solution:
For the sets to be Equal, both the sets should have the same elements
Yes, X and Y are Equal.
Yes, P and Q are Equal.
No, M and N are not Equal.
Q5: Identify Equivalent sets for the set R = {1, 2, 3}.
S = {a, b, c}
T = {4, 5, 6,8 }
U = {red, blue, green}
Solution:
Two Equal sets are said to be Equivalent if they have the same cardinal number. Hence,
Yes, S is Equivalent to R.
No, T is not Equivalent to R.
Yes, U is Equivalent to R.
Q6: Verify if the sets E = {apple, banana, cherry} and F = {guava, papaya, apple} are Equivalent.
Solution.
Yes, as both the sets contain three elements, they are Equivalent,
Q7: Determine if the sets G = {1, 5, 8} and H = {1, 2, 13, 14} are Equivalent or not.
Solution:
They are not Equivalent, as both have different numbers of elements.
Q9: Find an Equivalent set for the set J = {apple, banana, cherry}.
Solution:
Any set having three elements can be an Equivalent set to J. For example, Y= {a, e, u} or Z= {2,5,7}.}
Q10: Check if the following sets are Equal or not : A = {letters in the word TRIANGLE} and B = {letters in the word INTEGRAL}
Solution:
As both the words contain T, R, I, A, N, G, L, and E, they are Equal sets.
Frequently Asked Questions
How can one differentiate between Equal and Equivalent sets?
Equal sets have exactly the same elements, whereas Equivalent sets have the same number of elements.
Do Equivalent sets necessarily have the same elements?
No, it is not necessary for Equivalent sets to have the same elements.
What is cardinality?
It is the number of elements in a set and is used to determine if two sets are Equivalent or not.
How can we find out if a set is Equal?
If two sets have the same elements, then they are said to be Equal.
Can Equal sets have elements in a different order?
Yes, Equal sets can have elements arranged in different order.
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