Determine whether each statement is true or false: (a) {11, 13} ⊆ {11, 12, 13, 14, 15} (b) ∅ ⊂ {1, 5, 9} (c) {s, t, u} ⊈ {s, t, u} (d) {c, d, f, g} ⊂ {d, f}
Understand the Problem
The question presents four statements involving set theory and asks to determine whether each statement is true or false. The statements involve concepts like subsets and the empty set.
Answer
(a) True (b) True (c) False (d) False
Answer for screen readers
(a) True (b) True (c) False (d) False
Steps to Solve
- Evaluate statement (a): {11, 13} ⊆ {11, 12, 13, 14, 15}
Check if every element in the first set {11, 13} is also in the second set {11, 12, 13, 14, 15}. Since 11 and 13 are both present in the second set, the first set is a subset of the second set. Therefore, the statement is true.
- Evaluate statement (b): Ø ⊂ {1, 5, 9}
Check if the empty set Ø is a proper subset of {1, 5, 9}. The empty set is a subset of every set, and since {1, 5, 9} is not empty, the empty set is a proper subset of it. Therefore, the statement is true.
- Evaluate statement (c): {s, t, u} ⊈ {s, t, u}
Check if {s, t, u} is not a subset of {s, t, u}. The set {s, t, u} is a subset of itself. Since the statement says it is not a subset, the statement is false.
- Evaluate statement (d): {c, d, f, g} ⊂ {d, f}
Check if {c, d, f, g} is a proper subset of {d, f}. For {c, d, f, g} to be a subset of {d, f}, all its elements must be in {d, f}. However, 'c' and 'g' are in the first set but not in the second set, so {c, d, f, g} is not a subset of {d, f}. Thus, the statement is false.
(a) True (b) True (c) False (d) False
More Information
In set theory:
- A subset includes the possibility of being equal to the original set.
- A proper subset excludes the possibility of being equal to the original set.
- The empty set is a subset of every set.
Tips
A common mistake is confusing the subset symbol $⊆$ with the proper subset symbol $⊂$. $A ⊆ B$ means A is a subset of B (A can be equal to B). $A ⊂ B$ means A is a proper subset of B (A cannot be equal to B).
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