Q.Fill in the blanks to make each of the following a true statement :

Concept: Subset Listing — these are direct applications of complement definitions and set identities.

1. $A \cup A'$ is the set of all elements that are either in $A$ or not in $A$. That covers everything in the universal set $U$.
2. $\phi'$ is the complement of the empty set, which is $U$. So $U \cap A = A$. …

The Core Idea: What Complement Means

The complement of a set $A$, written $A'$, is everything in the universal set $U$ that is not in $A$. So $A$ and $A'$ are like two halves of a whole — they have no overlap, and together they make up the entire universe $U$.

This single idea — that $A$ and $A'$ are disjoint and exhaustive — is all you need to answer every part of this question. Let's apply it.

Step-by-Step Reasoning

1. (i) $A \cup A'$

The union of a set and its complement is the set of all elements that are either in $A$ or not in $A$. Since every element of the universal set $U$ is either in $A$ or in $A'$ (by definition of complement), this union must be the entire universal set.

So the blank is filled by $U$.

2. (ii) $\phi' \cap A$

First, what is $\phi'$? The complement of the empty set is the set of all elements not in $\phi$. Since $\phi$ contains nothing, everything in $U$ is "not in $\phi$". Therefore $\phi' = U$.

Now the expression becomes $U \cap A$. The intersection of the universal set with any set $A$ is just $A$ itself — because every element of $A$ is already in $U$, and $U$ adds nothing new.

Thus $\phi' \cap A = A$.

3. (iii) $A \cap A'$

A set and its complement have no elements in common — by definition, $A'$ contains only elements that are not in $A$. Their intersection is therefore empty. …

Here is the clear, concept-first solution for the given set theory problems.

Method: Set Theory Laws (Complement & Universal Set)

This method relies on understanding the definitions of the Universal Set ($U$) , the Complement ($A'$) , and the Empty Set ($\phi$) .

• Complement ($A'$) : All elements in $U$ that are not in $A$.
• Universal Set ($U$) : The set of all elements under consideration.
• Empty Set ($\phi$) : The set with no elements.

Steps to Solve

Step 1: Recall the core definitions.

• $A \cup A' = U$ (Union of a set and its complement gives the whole universe).
• $A \cap A' = \phi$ (Intersection of a set and its complement is empty).
• $\phi' = U$ (The complement of the empty set is the universal set).
• $U' = \phi$ (The complement of the universal set is the empty set).

Step 2: Apply the definitions to each blank.

(i) $A \cup A' = \dots$

• Why: Every element is either in $A$ or not in $A$. Together, they cover everything.
• Answer: $\boxed{U}$

(ii) $\phi' \cap A = \dots$

• Why: First, $\phi' = U$. So the expression becomes $U \cap A$. The intersection of the universal set with any set $A$ is just $A$.
• Answer: $\boxed{A}$ …

Here are the common mistakes students make with subset listing and set complement problems, specifically for the given fill-in-the-blanks, along with how to avoid each.

Mistake 1: Confusing the Complement (') with "Not in the Set" vs. "Everything Else"

The Mistake: Students often think $A'$ means "everything that is not in A" in an absolute sense, forgetting that it is always relative to the universal set $U$. This leads to errors in (i) and (iv).

• Example of error: For (i) $A \cup A'$, a student might think the answer is "everything" but write $\phi$ (empty set) or just "A" because they don't visualize the union.
• How to avoid: Always draw a Venn diagram with a rectangle for $U$. Shade $A$ in one color and $A'$ (everything outside $A$ but inside $U$) in another. The union of these two shaded regions is the entire rectangle.

Correct Answer (i): $U$ (the universal set).

Mistake 2: Forgetting that the Complement of the Universal Set ($U'$) is the Empty Set ($\phi$)

The Mistake: In (iv) $U' \cap A$, students treat $U'$ as "everything not in U" but then try to intersect it with A, leading to confusion.

• Example of error: A student might write $A$ or $\phi$ without reasoning. They forget that $U$ contains all elements under consideration, so nothing exists outside it.
• How to avoid: Memorize the two key complement facts:
  1. $U' = \phi$ (Nothing is outside the universe).
  2. $\phi' = U$ (The complement of nothing is everything). Then, apply the intersection: $\phi \cap A = \phi$.

Correct Answer (iv): $\phi$ (empty set).

Mistake 3: Misapplying the Intersection of a Set with its Complement

The Mistake: In (iii) $A \cap A'$, students confuse this with the union. They think the answer is $U$ (the union result) or they forget that a set and its complement have no common elements.

• Example of error: Writing $U$ instead of $\phi$.
• How to avoid: Use the definition: $A'$ contains everything not in $A$. Therefore, there is no element that can be both in $A$ and in $A'$ simultaneously. The intersection of two sets with no common elements is the empty set.

Correct Answer (iii): $\phi$ (empty set).

Mistake 4: Forgetting that the Complement of the Empty Set ($\phi'$) is $U$

The Mistake: In (ii) $\phi' \cap A$, students correctly find $\phi'$ but then forget how to intersect it with $A$. …