Question

If \(\mathrm{A}=\\{1,3,5,7,9,11,13,15,17\\}, \mathrm{B}=\\{2,4, \ldots 18\\}\) and \(\mathrm{N}\) is the universal set then \(\mathrm{A}^{\prime} \cup\left(\mathrm{A} \cup\left(\mathrm{B} \cap \mathrm{B}^{\prime}\right)\right.\) ) is (a) \(\mathrm{A}\) (b) B (c) \(A \cup B\) (d) \(\mathrm{N}\)

Short answer

The short answer to the question is: The expression \(A' \cup (A \cup (B \cap B'))\) is equal to \(\textbf{(d) } N.\)

Step-by-step solution

Find A' and B'

Since A and B are given, we can find A' and B' as the elements in the universal set N that are not in A and B. We are not given the exact elements of N, but we can infer that it is the set of all integers from 1 to 18 because set A contains odd integers and set B contains even integers in that range. A' = {2, 4, 6, 8, 10, 12, 14, 16, 18} B' = {1, 3, 5, 7, 9, 11, 13, 15, 17}

Calculate the intersection B ∩ B'

Since B and B' are complementary sets, their intersection B ∩ B' will be an empty set (∅), as no elements will be common in both B and B'. B ∩ B' = ∅

Calculate the union A ∪ (B ∩ B')

We calculated in step 2 that B ∩ B' is an empty set. The union of any set with an empty set is the set itself. Therefore, A ∪ (B ∩ B') = A ∪ ∅ = A = {1, 3, 5, 7, 9, 11, 13, 15, 17}

Calculate the expression A' ∪ (A ∪ (B ∩ B'))

In step 3, we found that A ∪ (B ∩ B') is equal to set A. Now we just need to calculate the union of A' with the set A: A' ∪ A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} Since this union contains all integers from 1 to 18, it seems to represent the universal set N. So the expression A' ∪ (A ∪ (B ∩ B')) is equal to N.

Match the result with the given options

Comparing our result with the given options, we find: (a) A (b) B (c) A ∪ B (d) N The expression A' ∪ (A ∪ (B ∩ B')) matches with option (d), which is N.

Key concepts

Universal Set

In set theory, a universal set is the set that contains all possible elements for a particular discussion. It serves as a "container" for all other sets that are being considered. Think of it as the biggest set that holds everything you might be dealing with.

In the given exercise, the universal set (denoted by \(N\)) is not explicitly listed, but we infer from context that it includes all integers from 1 to 18. This assumption is based on the elements in sets \(A\) and \(B\), where \(A\) has odd numbers and \(B\) has even numbers in that range. By encompassing elements from both \(A\) and \(B\), the universal set \(N\) ensures no numbers between 1 and 18 are left out.

Complement of a Set

The complement of a set is the collection of elements in the universal set that do not belong to the given set. It is denoted by adding a prime (') symbol after the set's name. This concept helps us understand what is outside a particular set but still within the bounds of the universal set.

For instance, if we have a set \(A\) and a universal set \(N\), the complement of \(A\) (denoted as \(A'\)) contains all elements that are in \(N\) but not in \(A\). From the provided exercise, since set \(A\) contains all odd numbers from 1 to 17, we can determine \(A'\) includes all even numbers from 2 to 18.

Intersection of Sets

The intersection of two sets consists of elements that are present in both sets. It is represented using the symbol \(\cap\). Think of the intersection as the overlap between two groups of items.

In the exercise, we look at the intersection \(B \cap B'\). Given \(B\) and \(B'\) are complements, they have no common elements. When two sets have nothing in common, their intersection is an empty set, denoted by \(\emptyset\). This is a key step, ensuring clarity in more complex operations involving intersections and their relation to other set calculations.

Union of Sets

The union of sets represents the combination of all elements from the involved sets. It is depicted with the \(\cup\) symbol. The union operation gathers everything, making sure no element is counted twice.

For example, to determine \(A \cup (B \cap B')\), since \(B \cap B'\) was an empty set, the union would simply be \(A\). An empty set doesn't add any new elements, so \(A \cup \emptyset = A\). Finally, for the full expression \(A' \cup (A \cup (B \cap B'))\), the union of \(A'\) (even numbers) and \(A\) (odd numbers) incorporates all numbers from 1 to 18, essentially forming the universal set \(N\). This union operation is vital for combining sets without losing any elements.