For each pair of statements, choose the one that is true: (a) 2 ⊆ {1, 2} or {2} ⊆ {1, 2} (b) p ∉ {q, r, s} or p ⊆ {p, q, r} (c) {6, 7, 8} ⊆ {1, 2, 3, 4, ...} or {6, 7, 8} ∈ {1, 2,... For each pair of statements, choose the one that is true: (a) 2 ⊆ {1, 2} or {2} ⊆ {1, 2} (b) p ∉ {q, r, s} or p ⊆ {p, q, r} (c) {6, 7, 8} ⊆ {1, 2, 3, 4, ...} or {6, 7, 8} ∈ {1, 2, 3, 4, ...} (d) x ∈ {w, x} or {x} ∈ {w, x}
Understand the Problem
The question presents four pairs of statements involving sets and asks you to identify the true statement within each pair. The statements involve understanding set theory concepts such as subsets (⊆), elements (∈), and the difference between an element and a set containing that element.
Answer
(a) {2} ⊆ {1, 2} (b) p ∉ {q,r,s} (c) {6, 7, 8} ⊆ {1, 2, 3, 4, ...} (d) x ∈ {w, x}
Answer for screen readers
(a) {2} ⊆ {1, 2} (b) p ∉ {q,r,s} (c) {6, 7, 8} ⊆ {1, 2, 3, 4, ...} (d) x ∈ {w, x}
Steps to Solve
- Analyze option (a)
2 ⊆ {1,2} is false because the subset symbol ⊆ is used to relate two sets. Here, 2 is an element, not a set. {2} ⊆ {1, 2} is true because the set containing only the element 2 is a subset of the set containing 1 and 2.
- Analyze option (b)
p ∉ {q,r,s} is false. We don't know if p is not an element of {q, r, s}. p ⊆ {p,q,r} is false. The subset symbol ⊆ is used to relate two sets. Here, p is an element, not a set.
- Analyze option (c)
{6, 7, 8} ⊆ {1, 2, 3, 4, ...} is true because all elements in the set {6, 7, 8} are also elements of the infinite set {1, 2, 3, 4, ...}. {6, 7, 8} ∈ {1, 2, 3, 4, ...} is false because {6,7,8} is not an element of the set {1, 2, 3, 4, ...}. The elements of the set {1, 2, 3, 4, ...} are individual numbers, not sets of numbers.
- Analyze option (d)
x ∈ {w, x} is true because x is an element of the set {w, x}. {x} ∈ {w, x} is false because the set {x} is not an element of the set {w, x}. The elements of {w, x} are w and x.
(a) {2} ⊆ {1, 2} (b) p ∉ {q,r,s} (c) {6, 7, 8} ⊆ {1, 2, 3, 4, ...} (d) x ∈ {w, x}
More Information
Understanding the difference between an element and a set is crucial in set theory. Always remember that an element is a member of a set, while a subset is a set contained within another set.
Tips
A common mistake is confusing the element symbol ∈ with the subset symbol ⊆. Remember that ∈ is used to indicate that an element belongs to a set, while ⊆ indicates that a set is a subset of another set.
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