Question:

Let A= {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?
(i) {3, 4} \(⊂\) A
(ii) {3, 4}}\(∈\) A
(iii) {{3, 4}}\(⊂\) A
(iv) 1\(∈\) A
(v) 1\(⊂\) A
(vi) {1, 2, 5} \(⊂\) A
(vii) {1, 2, 5}\(∈\) A
(viii) {1, 2, 3} \(⊂\) A
(ix) \( \phi ∈A\)
(x) \( \phi⊂ A\)
(xi)  {\(\phi\)}\( ⊂ A\)

Updated On: Oct 22, 2023
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Solution and Explanation

A = {1, 2, {3, 4}, 5}
(i) The statement {3, 4} \(⊂ A\) is incorrect because 3\(∈\) {3, 4}; however, \(3 ∉ A.\)


(ii) The statement {3, 4}\(∈ A\) is correct because {3, 4} is an element of A.


(iii) The statement {{3, 4}} \(⊂ A\) is correct because {3, 4} \(∈\) {{3, 4}} and {3, 4}\(∈ A.\)


(iv) The statement 1 \(∈ A\)  is correct because 1 is an element of A.


(v) The statement 1\(⊂ A\) is incorrect because an element of a set can never be a subset of itself.


(vi) The statement {1, 2, 5} \(⊂ A\) is correct because each element of {1, 2, 5} is also an element of A.


(vii) The statement {1, 2, 5} \(∈ A\)  is incorrect because {1, 2, 5} is not an element of A.


(viii) The statement {1, 2, 3} \(⊂ A\)  is incorrect because \(3 ∈\) {1, 2, 3}; however, 3 A.


(ix) The statement \(\phi∈ A\)  is incorrect because \(\phi\) is not an element of A.


(x) The statement \(\phi ⊂ A\)  is correct because \(\phi\)  is a subset of every set.


(xi) The statement {\(\phi\)\(⊂ A\) is incorrect because \(\phi ∈\) {\(\phi\)}; however, \(\phi∈ A.\)

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Concepts Used:

Types of Sets

Sets are of various types depending on their features. They are as follows:

  • Empty Set - It is a set that has no element in it. It is also called a null or void set and is denoted by Φ or {}.
  • Singleton Set - It is a set that contains only one element.
  • Finite Set - A set that has a finite number of elements in it.
  • Infinite Set - A set that has an infinite number of elements in it.
  • Equal Set - Sets in which elements of one set are similar to elements of another set. The sequence of elements can be any but the same elements exist in both sets.
  • Sub Set - Set X will be a subset of Y if all the elements of set X are the same as the element of set Y.
  • Power Set - It is the collection of all subsets of a set X.
  • Universal Set - A basic set that has all the elements of other sets and forms the base for all other sets.
  • Disjoint Set - If there is no common element between two sets, i.e if there is no element of Set A present in Set B and vice versa, then they are called disjoint sets.
  • Overlapping Set - It is the set of two sets that have at least one common element, called overlapping sets.