Q.If $A = \{1, 2, 3\}$, $B = \{3, 4\}$ and $C = \{4, 5, 6\}$, then $(A \times B) \cup (A \times C) = \{(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)\}$.

The key idea here is the distributive property of the Cartesian product over set union.

1. First, determine the set $B \cup C$.

   $B \cup C = \{3, 4\} \cup \{4, 5, 6\} = \{3, 4, 5, 6\}$.

2. Next, calculate the Cartesian product $A \times (B \cup C)$.

   $A \times (B \cup C) = \{1, 2, 3\} \times \{3, 4, 5, 6\}$

   $= \{(1, 3), (1, 4), (1, 5), (1, 6),$ …

The core concept here is the Cartesian product of sets, which creates ordered pairs, and the union of sets, which combines elements. We are asked to evaluate an expression involving both operations and compare it to a given set.

A key property that simplifies such problems is the distributive law of Cartesian products over set union. This property states that for any sets $A$, $B$, and $C$:

This means we can either first find the union of $B$ and $C$ and then take its Cartesian product with $A$, or we can find the Cartesian products $A \times B$ and $A \times C$ separately and then take their union. Both approaches will yield the same result. We will demonstrate this by calculating both sides of the equation.

Let's break down the calculation step-by-step.

1. Identify the given sets:

   We are given three sets:

   $A = \{1, 2, 3\}$

   $B = \{3, 4\}$

   $C = \{4, 5, 6\}$

2. Calculate $A \times B$:

   The Cartesian product $A \times B$ consists of all possible ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

   For each element in $A$, we pair it with every element in $B$:

   • For $1 \in A$: $(1, 3), (1, 4)$
   • For $2 \in A$: $(2, 3), (2, 4)$
   • For $3 \in A$: $(3, 3), (3, 4)$ Combining these, we get: $A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)\}$

3. Calculate $A \times C$:

   Similarly, $A \times C$ consists of all possible ordered pairs $(a, c)$ where $a \in A$ and $c \in C$.

   • For $1 \in A$: $(1, 4), (1, 5), (1, 6)$
   • For $2 \in A$: $(2, 4), (2, 5), (2, 6)$
   • For $3 \in A$: $(3, 4), (3, 5), (3, 6)$ Combining these, we get: $A \times C = \{(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)\}$

4. Calculate $(A \times B) \cup (A \times C)$:

   The union of two sets combines all unique elements from both sets. In this case, the elements are ordered pairs. We list all pairs from $A \times B$ and $A \times C$, ensuring we do not repeat any pair that appears in both sets.

   Pairs from $A \times B$: $\{(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)\}$

   Pairs from $A \times C$: $\{(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)\}$

   Notice that the pairs $(1, 4)$, $(2, 4)$, and $(3, 4)$ are present in both $A \times B$ and $A \times C$. When forming the union, we include them only once. …