(A ∪ B) is equal to (a) (A ∩ B) (c) A` ∩ B` (b) A ∪ B` (d) none of the...
**Explanation:**
To understand the relationship between the sets (A ∪ B) and the options provided, we need to recall the definitions of the operations involved:
- **Union (A ∪ B):** The union of sets A and B is the set that contains all the elements that are in either A or B, or in both. In other words, it is the combination of all the elements from both sets.
Now let's analyze each option in detail:
**Option (a): (A ∩ B)**
- **Explanation:** The intersection of sets A and B represents the set of elements that are common to both sets. This option does not represent the union of sets A and B, but rather their intersection. Therefore, option (a) is not equal to (A ∪ B).
**Option (b): A ∪ B`**
- **Explanation:** The complement of set B (denoted as B`) represents all the elements that are not in set B. The union of set A and the complement of set B represents the set of elements that are in either A or not in B. Therefore, option (b) is equivalent to (A ∪ B`).
**Option (c): A` ∩ B`**
- **Explanation:** The complement of set A (denoted as A`) represents all the elements that are not in set A. The intersection of the complements of sets A and B represents the set of elements that are not in A and not in B. This option does not represent the union of sets A and B, but rather the intersection of their complements. Therefore, option (c) is not equal to (A ∪ B).
**Option (d): None of these**
- **Explanation:** As we have determined, option (b) is equal to (A ∪ B`). Therefore, option (d), which states that none of the provided options are correct, is incorrect.
In conclusion, the correct answer is **Option (b): A ∪ B`**, as it represents the union of sets A and the complement of set B.
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