Let A = {2,{4,5},4}. Which of the following statements are incorrect and why? Give an explanation for your answer to each part.
i. 5 ∈ A?
ii. {5} ∈ A?
iii. 5 ⊂ A?
Let C = {4,5}
Then A = {2,C,4}
They are all incorrect, and it should be clear why. What are your reasons?
If C={4,5} then {4,5} is treated as one element comprised of the separate elements 4 and 5 . Therefore, 5 itself is not an element of A . Thus, {5} cannot be an element of A. 5 is not a subset of A because it is a part of the pair of elements in C.
To determine whether each statement is correct or not, we need to understand the elements of set A.
Set A contains three elements:
1. The number 2
2. The set containing the numbers 4 and 5: {4, 5}
3. The number 4
Let's evaluate each statement:
i. 5 ∈ A?
This statement asks whether the number 5 is an element of set A. In this case, the statement is incorrect because set A does not contain the number 5. The only elements in A are 2, {4, 5}, and 4. To check if an element is in a set, we look for an exact match.
ii. {5} ∈ A?
This statement asks whether the set {5} is an element of set A. In this case, the statement is also incorrect. Although set A does contain the set {4, 5}, it does not contain the set {5}. This is because {5} is a different set from {4, 5}. Sets are considered distinct if their elements are not exactly the same.
iii. 5 ⊂ A?
This statement asks whether the number 5 is a proper subset of set A. A proper subset means that all elements of the subset are also elements of the set, but the subset is not equal to the set itself. In this case, the statement is incorrect. Set A does not contain the number 5 at all, so 5 cannot be considered as a subset of A.
To summarize:
- The statement "5 ∈ A?" is incorrect because 5 is not an element of set A.
- The statement "{5} ∈ A?" is incorrect because the set {5} is not an element of A.
- The statement "5 ⊂ A?" is incorrect because 5 is not a subset of A.