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Check, whether the following statements ...

Check, whether the following statements are true or false :
(i) `{1,2} cancel(sub){1,2,3}`
(ii) `{a,b}sub{x:x " is a letter of English alphabet"}`
(iii) `{x:x " is an odd natural number"}sube {x:x " is a positve integer"}`
(iv) `{a} in {a,b}`.

Text Solution

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The correct Answer is:
To determine whether the given statements are true or false, we will analyze each statement step by step. ### Statement (i): `{1,2} ⊄ {1,2,3}` 1. **Identify the Sets**: The first set is `{1, 2}` and the second set is `{1, 2, 3}`. 2. **Check Subset Condition**: A set A is a subset of set B if every element of A is also an element of B. 3. **Comparison**: The elements of `{1, 2}` are 1 and 2. Both of these elements are present in `{1, 2, 3}`. 4. **Conclusion**: Since all elements of the first set are in the second set, `{1, 2}` is indeed a subset of `{1, 2, 3}`. Therefore, the statement is **False**. ### Statement (ii): `{a,b} ⊆ {x : x " is a letter of English alphabet"}` 1. **Identify the Sets**: The first set is `{a, b}` and the second set consists of all letters of the English alphabet. 2. **Check Subset Condition**: We need to see if both `a` and `b` are letters of the English alphabet. 3. **Comparison**: Both `a` and `b` are indeed letters of the English alphabet. 4. **Conclusion**: Since both elements of the first set are in the second set, the statement is **True**. ### Statement (iii): `{x : x " is an odd natural number"} ⊆ {x : x " is a positive integer"}` 1. **Identify the Sets**: The first set contains all odd natural numbers (1, 3, 5, ...), and the second set contains all positive integers (1, 2, 3, 4, ...). 2. **Check Subset Condition**: We need to see if every odd natural number is also a positive integer. 3. **Comparison**: All odd natural numbers (1, 3, 5, ...) are indeed positive integers. 4. **Conclusion**: Therefore, the statement is **True**. ### Statement (iv): `{a} ∈ {a, b}` 1. **Identify the Sets**: The first set is `{a}` and the second set is `{a, b}`. 2. **Check Membership Condition**: We need to determine if the set `{a}` is an element of the set `{a, b}`. 3. **Comparison**: The set `{a, b}` contains the elements `a` and `b`, but it does not contain the set `{a}` as an element. 4. **Conclusion**: Therefore, the statement is **False**. ### Final Results: - (i) False - (ii) True - (iii) True - (iv) False
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