Let A = {1, 2, 3}. Which of the following statements is correct?
A{2} ∈ A, because 2 is an element of A
B2 ⊆ A, because 2 is contained within A
C{2} ⊆ A, because every element of {2} is also an element of A
D{2} and 2 are the same object, so both ∈ and ⊆ apply to either
The element 2 and the set {2} are different kinds of objects. The number 2 is an element of A (2 ∈ A), but you cannot write 2 ⊆ A because 2 is not a set. The set {2} is not an element of A — A contains the number 2, not a set containing 2 — so {2} ∉ A. But {2} ⊆ A is true: every member of {2}, which is just the number 2, is also in A. The ∈/⊆ distinction is not stylistic; it tracks a fundamental type difference between elements and sets.
Question 2 Multiple Choice
To prove that two sets A and B are equal, the standard mathematical approach is:
AShow that they have the same number of elements and that each element of A matches one in B
BShow that A ⊆ B and B ⊆ A (double containment)
CShow that every element of A is an element of B
DShow that A ∪ B = A and A ∩ B = B
Double containment (A ⊆ B and B ⊆ A) is the standard proof strategy for set equality. Showing A ⊆ B alone (option C) only proves containment, not equality — B could still have extra elements. Showing the same number of elements (option A) works for finite sets but is not the general method and says nothing about which elements are present. The double containment argument works for all sets, finite or infinite, and directly uses the definition of the subset relation.
Question 3 True / False
The empty set ∅ is not a subset of any non-empty set, because it shares no elements with it.
TTrue
FFalse
Answer: False
This is a very common misconception. The subset relation A ⊆ B means: for all x, if x ∈ A then x ∈ B. For A = ∅, there are no elements to check — the condition 'if x ∈ ∅ then x ∈ B' is vacuously true for any B, because the hypothesis 'x ∈ ∅' is never satisfied. Therefore ∅ ⊆ B for every set B, including non-empty sets. Having no elements in common is not the definition of 'not a subset'; that's a description of disjointness.
Question 4 True / False
If x ∈ A and A ⊆ B, then x ∈ B.
TTrue
FFalse
Answer: True
This follows directly from the definition of ⊆. A ⊆ B means: for all z, if z ∈ A then z ∈ B. Applying this universal statement to the specific element x: since x ∈ A, we conclude x ∈ B. This chain of reasoning — element of a set, which is a subset of another, so element of the larger set — is one of the most frequently used inference patterns in formal proofs involving sets.
Question 5 Short Answer
Explain the difference between writing 3 ∈ A and {3} ⊆ A. Are these two statements equivalent? What are 3 and {3} respectively?
Think about your answer, then reveal below.
Model answer: 3 is a number (an element); {3} is a set containing the number 3. '3 ∈ A' says the number 3 is a member of A. '{3} ⊆ A' says the set {3} is contained in A — equivalently, every element of {3} (which is just 3) belongs to A. If A contains the number 3, both statements are true simultaneously, but they assert different things about different objects. They are not equivalent: '3 ∈ A' talks about an element, '{3} ⊆ A' talks about a set.
This distinction is the source of the most persistent errors in formal set theory. Writing '{3} ∈ A' would mean {3} is itself a member of A — which requires A to be a set of sets, like A = {{3}, {4}}. Conflating ∈ and ⊆ breaks proofs because the two symbols operate on different types of objects: ∈ relates an element to a set, while ⊆ relates a set to a set.