To solve the problem from the provided text:

Problem Analysis:

Given sets $A = \{1, 2, 5\}$ and $B = \{0, 4\}$, we define a relation $R$ from $A$ to $B$, where $(x, y) \in R$ if and only if $x - y$ is even.

Step-by-Step Solution:

1. Determine Cartesian Product $A \times B$: $A \times B = \{(x, y) \mid x \in A, y \in B\}$ Thus: $A \times B = \{(1, 0), (1, 4), (2, 0), (2, 4), (5, 0), (5, 4)\}$

2. Apply the condition $x - y$ is even:

   • Compute $x - y$ for each pair in $A \times B$:
     • $(1, 0): 1 - 0 = 1$ (odd, not in $R$).
     • $(1, 4): 1 - 4 = -3$ (odd, not in $R$).
     • $(2, 0): 2 - 0 = 2$ (even, in $R$).
     • $(2, 4): 2 - 4 = -2$ (even, in $R$).
     • $(5, 0): 5 - 0 = 5$ (odd, not in $R$).
     • $(5, 4): 5 - 4 = 1$ (odd, not in $R$).

3. Identify the relation $R$: The pairs that satisfy the condition are: $R = \{(2, 0), (2, 4)\}$

Final Answer:

• The elements of $A \times B$ are: $\{(1, 0), (1, 4), (2, 0), (2, 4), (5, 0), (5, 4)\}.$
• The elements of $R$ are: $\{(2, 0), (2, 4)\}.$

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Related Questions:

1. How do we define a relation between two sets formally?
2. What is the difference between even and odd numbers in modular arithmetic?
3. Can $R$ form a function if every $x \in A$ maps to exactly one $y \in B$?
4. How does the Cartesian product $A \times B$ relate to relations?
5. What happens if we change the condition to $x + y$ being even?

Tip:

When working with relations, always check if the condition imposes symmetry, reflexivity, or transitivity properties!