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Wednesday, 19 March 2014

Equal Sets and Equivalent Sets

Consider two sets:
A = {−9, −3, 0, 5, 12}
B = {−2, 1, 2, 4, 7}
Did you notice any relation between the sets and B?
Let us see.
We have,
A = {−9, −3, 0, 5, 12}
B = {−2, 1, 2, 4, 7}
Therefore, we have n (A) = 5 and n (B) = 5
Observe that both the sets A and B have same number of elements. Therefore, in this case, we say that the sets A and B are equivalent sets and it can be defined as:
Two finite sets are called equivalent, if they have the same number of elements. 
Thus, two finite sets X and Y are equivalent, if n (X) = n (Y). We write it as  Y (read as “X is equivalent to Y”)
Now, consider the two sets:
X = {all letters in the word STONE}
Y = {all letters in the word NOTES}
Did you notice any relation between the sets and Y?
Let us see.
We have,
X = {S, T, O, N, E} and Y = {N, O, T, E, S}
Observe that both the sets X and Y have same elements. Therefore, in this case, we say that the sets X andY are equal sets.
Two sets are called equal, if they have same elements.
When two sets X and Y are equal, we denote it as X = Y; and if they are not equal, then we write it as X Y
Also, note that n (X) = 5 and n (Y) = 5
Therefore, we can conclude that:
If A and B are finite sets and A = B, then (A) = n (B) i.e., A and B are equivalent. However, the converse of the above statement may not be true.
For example, if A = {2, 4, 6} and B = {1, 3, 5}, then n (A) = n (B) = 3; however, A  B
Let us now look at some more examples to understand the above discussed concepts better.
Example 1:
Which of the following sets are equal?
(a) X = {xx is a letter in the word REFRESH},
Y = {A letter in the word FRESHER}
(b) X = {4}, Y = {xx∈ N, x − 4 = 0}
(d) X = {x/x is a vowel letter in the word WEIGHT},
Y = {x/x is a vowel letter in the word HEIGHT}
(c) X = {xx∈ N, 0 < x < 4}, Y = {xx∈ W}
Solution:
(a) X = {R, E, F, S, H}, Y = {F, R, E, S, H}
∴ X and Y are equal sets i.e., X = Y
(b) X = {4}, Y = {4}
∴ X and Y are equal sets i.e., X = Y
(c) X = {E, 1}, Y = {E, I}
∴ X and Y are equal sets i.e., X = Y
(d) X = {1, 2, 3}, Y = {0, 1, 2, 3, 4, 5 …}
∴ X and Y are not equal sets i.e., X  Y
Example 2:
Which of the following sets are equal?
(a) X = {xx is a vowel in the word MATRIX}
Y = {A vowel in the word SHIVANI}
(b) X = {5, 0, 1, 0, 2, 1}, Y = {3, 2, 8, 3, 3, 2}
Solution:
(a) X = {A, I}, Y = {A, I}
∴ n (X) = 2 and n (Y) = 2
This means n (X) = n (Y)
Therefore, X and Y are equivalent sets.
(b) X = {5, 0, 1, 0, 2, 1} = {0, 1, 2, 5}, Y = {3, 2, 8, 3, 3, 2} = {2, 3, 8}
∴ n (X) = 4 and n (Y) = 3
This means n (X) ≠ n (Y)
Therefore, X and Y are not equivalent sets.
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