Determine Truth Value of Statements on Relation Definition

To determine the truth value of the given statements regarding the relation $R$ defined on the set $A = \{-2, -1, 0, 1, 2\}$, we analyze the condition for membership in $R$: $(a, b) \in R \iff 1 + ab > 0 \iff ab > -1$

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Analysis of Statement I: The number of elements in $R$ is 17

We count the number of ordered pairs $(a, b) \in A \times A$ that satisfy $ab > -1$ by checking each possible value of $a \in A$:

1. For $a = -2$:

   • $b = -2 \implies ab = 4 > -1$ $\quad\rightarrow\quad (-2, -2) \in R$
   • $b = -1 \implies ab = 2 > -1$ $\quad\rightarrow\quad (-2, -1) \in R$
   • $b = 0 \implies ab = 0 > -1$ $\quad\rightarrow\quad (-2, 0) \in R$
   • $b = 1 \implies ab = -2 \ngtr -1$
   • $b = 2 \implies ab = -4 \ngtr -1$

   (3 elements: $(-2, -2), (-2, -1), (-2, 0)$)

2. For $a = -1$:

   • $b = -2 \implies ab = 2 > -1$ $\quad\rightarrow\quad (-1, -2) \in R$
   • $b = -1 \implies ab = 1 > -1$ $\quad\rightarrow\quad (-1, -1) \in R$
   • $b = 0 \implies ab = 0 > -1$ $\quad\rightarrow\quad (-1, 0) \in R$
   • $b = 1 \implies ab = -1 \ngtr -1$
   • $b = 2 \implies ab = -2 \ngtr -1$

   (3 elements: $(-1, -2), (-1, -1), (-1, 0)$)

3. For $a = 0$:

   • $b = -2, -1, 0, 1, 2 \implies ab = 0 > -1$ for all $b \in A$.

   (5 elements: $(0, -2), (0, -1), (0, 0), (0, 1), (0, 2)$)

4. For $a = 1$:

   • $b = -2 \implies ab = -2 \ngtr -1$
   • $b = -1 \implies ab = -1 \ngtr -1$
   • $b = 0 \implies ab = 0 > -1$ $\quad\rightarrow\quad (1, 0) \in R$
   • $b = 1 \implies ab = 1 > -1$ $\quad\rightarrow\quad (1, 1) \in R$
   • $b = 2 \implies ab = 2 > -1$ $\quad\rightarrow\quad (1, 2) \in R$

   (3 elements: $(1, 0), (1, 1), (1, 2)$)

5. For $a = 2$:

   • $b = -2 \implies ab = -4 \ngtr -1$
   • $b = -1 \implies ab = -2 \ngtr -1$
   • $b = 0 \implies ab = 0 > -1$ $\quad\rightarrow\quad (2, 0) \in R$
   • $b = 1 \implies ab = 2 > -1$ $\quad\rightarrow\quad (2, 1) \in R$
   • $b = 2 \implies ab = 4 > -1$ $\quad\rightarrow\quad (2, 2) \in R$

   (3 elements: $(2, 0), (2, 1), (2, 2)$)

Summing the number of valid pairs: $n(R) = 3 + 3 + 5 + 3 + 3 = 17$

Thus, Statement I is True.

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Analysis of Statement II: $R$ is an equivalence relation

An equivalence relation must be reflexive, symmetric, and transitive.

1. Reflexivity: For any $a \in A$, $1 + a^2 \ge 1 > 0$, so $(a, a) \in R$ for all $a \in A$. Hence, $R$ is reflexive.

2. Symmetry: If $(a, b) \in R$, then $1 + ab > 0 \implies 1 + ba > 0 \implies (b, a) \in R$. Hence, $R$ is symmetric.

3. Transitivity: We check for transitivity. Consider $a = -2$, $b = 0$, and $c = 2$:

   • $1 + a \cdot b = 1 + (-2)(0) = 1 > 0 \implies (-2, 0) \in R$
   • $1 + b \cdot c = 1 + (0)(2) = 1 > 0 \implies (0, 2) \in R$
   • $1 + a \cdot c = 1 + (-2)(2) = -3 \ngtr 0 \implies (-2, 2) \notin R$

   Since $(-2, 0) \in R$ and $(0, 2) \in R$, but $(-2, 2) \notin R$, the relation $R$ is not transitive.

Thus, $R$ is not an equivalence relation, making Statement II False.

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Conclusion:

• Statement I is true.
• Statement II is false.

Hence, the correct option is A (Only I is true).