Let's solve the problem presented.

Given:

• $A = \{0, 1, 2, 3\}$
• $B = \{-3, -1, 3, 6\}$

We are asked to form two relations $R_1$ and $R_2$ from set $A$ to set $B$ based on the specified conditions.

a. Relation $R_1 = \{(a, b) | ab > 0; a \in A, b \in B\}$

This relation consists of pairs $(a, b)$ where the product $ab$ is greater than zero. Let's evaluate each possible pair:

• For $a = 0$: $0 \times b = 0$ for all $b \in B$. So, no pairs with $a = 0$ will satisfy $ab > 0$.
• For $a = 1$:
  • $1 \times (-3) = -3$ (not > 0)
  • $1 \times (-1) = -1$ (not > 0)
  • $1 \times 3 = 3$ (satisfies $ab > 0$)
  • $1 \times 6 = 6$ (satisfies $ab > 0$)
  • Thus, pairs are $(1, 3)$ and $(1, 6)$.
• For $a = 2$:
  • $2 \times (-3) = -6$ (not > 0)
  • $2 \times (-1) = -2$ (not > 0)
  • $2 \times 3 = 6$ (satisfies $ab > 0$)
  • $2 \times 6 = 12$ (satisfies $ab > 0$)
  • Thus, pairs are $(2, 3)$ and $(2, 6)$.
• For $a = 3$:
  • $3 \times (-3) = -9$ (not > 0)
  • $3 \times (-1) = -3$ (not > 0)
  • $3 \times 3 = 9$ (satisfies $ab > 0$)
  • $3 \times 6 = 18$ (satisfies $ab > 0$)
  • Thus, pairs are $(3, 3)$ and $(3, 6)$.

So, the relation $R_1$ is: $R_1 = \{(1, 3), (1, 6), (2, 3), (2, 6), (3, 3), (3, 6)\}$

b. Relation $R_2 = \{(a, b) | a \leq b; a \in A, b \in B\}$

This relation consists of pairs $(a, b)$ where $a \leq b$.

• For $a = 0$:
  • $0 \leq -3$ (false)
  • $0 \leq -1$ (false)
  • $0 \leq 3$ (true, so $(0, 3)$)
  • $0 \leq 6$ (true, so $(0, 6)$)
• For $a = 1$:
  • $1 \leq -3$ (false)
  • $1 \leq -1$ (false)
  • $1 \leq 3$ (true, so $(1, 3)$)
  • $1 \leq 6$ (true, so $(1, 6)$)
• For $a = 2$:
  • $2 \leq -3$ (false)
  • $2 \leq -1$ (false)
  • $2 \leq 3$ (true, so $(2, 3)$)
  • $2 \leq 6$ (true, so $(2, 6)$)
• For $a = 3$:
  • $3 \leq -3$ (false)
  • $3 \leq -1$ (false)
  • $3 \leq 3$ (true, so $(3, 3)$)
  • $3 \leq 6$ (true, so $(3, 6)$)

So, the relation $R_2$ is: $R_2 = \{(0, 3), (0, 6), (1, 3), (1, 6), (2, 3), (2, 6), (3, 3), (3, 6)\}$

Would you like further details on how these pairs were derived?

Here are some related questions for further exploration:

1. How would these relations change if sets $A$ and $B$ had different elements?
2. Can you create a new relation using a different rule for pairs?
3. What happens if we restrict the elements in $B$ to positive values only?
4. How can we represent these relations visually using a Cartesian plane?
5. What are some real-life examples where such relations could apply?

Tip: When forming relations between sets, always test each possible pair to see if it satisfies the given condition.