Let's analyze the conditions one by one for the relation $R$ from set $A = \{0,1,2,3,4\}$ to set $B = \{0,1,2,3\}$.

(a) $a = b$

The ordered pairs where $a = b$ are simply those where the elements of $A$ and $B$ are equal: $R = \{(0,0), (1,1), (2,2), (3,3)\}$

(b) $a + b = 4$

The ordered pairs where the sum of $a$ and $b$ equals 4: $R = \{(0,4), (1,3), (2,2), (3,1), (4,0)\}$ However, note that $b$ must be in $B = \{0,1,2,3\}$, so exclude pairs with $b = 4$. Thus, the relation is: $R = \{(1,3), (2,2), (3,1), (4,0)\}$

(c) $a > b$

The ordered pairs where $a$ is greater than $b$: $R = \{(1,0), (2,0), (2,1), (3,0), (3,1), (3,2), (4,0), (4,1), (4,2), (4,3)\}$

(d) $a \mid b$

The ordered pairs where $a$ divides $b$ (meaning $b$ is a multiple of $a$): $R = \{(1,1), (1,2), (1,3), (2,2), (0,0)\}$ Note that $0$ divides $0$ but is typically not considered a divisor for non-zero numbers.

(e) $\gcd(a, b) = 1$

The ordered pairs where the greatest common divisor (gcd) of $a$ and $b$ is 1: $R = \{(1,0), (1,2), (1,3), (2,3), (3,2), (3,4)\}$ For $gcd(0, b)$ where $b \neq 0$, the gcd is $b$, so these are not included.

(f) $\text{lcm}(a, b) = 2$

The ordered pairs where the least common multiple (lcm) of $a$ and $b$ is 2: $R = \{(1,2), (2,1)\}$ Here $(1,2)$ and $(2,1)$ are the only pairs that give an lcm of 2.

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Related Questions:

1. How is the gcd of two numbers calculated?
2. What is the significance of lcm in number theory?
3. Can you explain the division algorithm used in condition (d)?
4. What are some real-world applications of relations in set theory?
5. How can we generalize these conditions for larger sets?

Tip: When working with relations, ensure you clearly understand the definitions of operations like gcd, lcm, and divisibility, as they often appear in various mathematical contexts.