Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
{3, 4} ⊂ A
Incorrect - {3, 4} is an element of A, not a subset. For {3, 4} ⊂ A to be true, elements 3 and 4 must belong to A, but they don't.
{3, 4} ∈ A
Correct - {3, 4} is listed as an element in A = {1, 2, {3, 4}, 5}.
{{3, 4}} ⊂ A
Correct - The set {{3, 4}} has only one element {3, 4}, which is an element of A.
1 ∈ A
Correct - 1 is explicitly listed as an element of A.
1 ⊂ A
Incorrect - 1 is an element of A, not a subset. The symbol ⊂ is used between sets, not between an element and a set.
{1, 2, 5} ⊂ A
Correct - All elements 1, 2, 5 belong to A, so {1, 2, 5} is a subset of A.
{1, 2, 5} ∈ A
Incorrect - {1, 2, 5} is not an element of A. The elements of A are 1, 2, {3, 4}, and 5 only.
{1, 2, 3} ⊂ A
Incorrect - {1, 2, 3} ⊂ A is false because 3 ∉ A. Elements 1 and 2 are in A, but 3 is not.
φ ∈ A
Incorrect - φ is not an element of A. The elements of A are 1, 2, {3, 4}, and 5.
φ ⊂ A
Correct - The empty set φ is a subset of every set, including A.
{φ} ⊂ A
Incorrect - {φ} ⊂ A would require φ ∈ A, but φ is not an element of A.
Explanation
This question tests understanding of the distinction between 'element of' (∈) and 'subset of' (⊂). The set A = {1, 2, {3, 4}, 5} has four elements: 1, 2, {3, 4}, and 5. Note that {3, 4} is a single element (a set within a set), not separate elements 3 and 4. For subset relation X ⊂ A, every element of X must be an element of A. For element relation X ∈ A, X must be one of the listed elements of A. The empty set φ is always a subset of any set, but it is not necessarily an element.