1. **State the problem:** We have three sets defined with conditions on $x$ and three values a=5, b=8, c=4. We want to find which option (A, B, C, or D) correctly matches the values to the sets. 2. **Analyze each set:** - Set I: $S = \{x \in \mathbb{Z} \mid -2 \leq x \leq 5\}$ means $x$ is an integer between -2 and 5 inclusive. - Set II: $T = \{x \mid \frac{x}{2} \in \mathbb{N}, 0 < x \leq 8\}$ means $x$ is such that $x/2$ is a natural number and $x$ is between 1 and 8 inclusive. - Set III: $P = \{(x+3) \in \mathbb{Z} \mid -7 < x < -1\}$ means $x+3$ is an integer and $x$ is strictly between -7 and -1. 3. **Evaluate each value for each set:** - For I: - a=5: Is 5 an integer between -2 and 5? Yes. - b=8: Is 8 an integer between -2 and 5? No. - c=4: Is 4 an integer between -2 and 5? Yes. - For II: - a=5: Is $5/2=2.5$ a natural number? No. - b=8: Is $8/2=4$ a natural number? Yes. - c=4: Is $4/2=2$ a natural number? Yes. - For III: - a=5: Check if $x+3=5$ with $-7 < x < -1$. Then $x=5-3=2$, but $2$ is not between -7 and -1, so no. - b=8: $x+3=8 \Rightarrow x=5$, not between -7 and -1, no. - c=4: $x+3=4 \Rightarrow x=1$, not between -7 and -1, no. We must check the values of $x+3$ for $x$ in $(-7,-1)$: Since $x$ is integer between -6 and -2 (because $x$ must be integer for $x+3$ to be integer), possible $x$ are -6, -5, -4, -3, -2. Then $x+3$ are: - $-6+3 = -3$ - $-5+3 = -2$ - $-4+3 = -1$ - $-3+3 = 0$ - $-2+3 = 1$ So $P = \{-3, -2, -1, 0, 1\}$. Check which of a=5, b=8, c=4 are in $P$: None of 5, 8, or 4 are in $P$. 4. **Conclusion:** - For I, valid values are a=5 and c=4. - For II, valid values are b=8 and c=4. - For III, none of a=5, b=8, c=4 are in $P$. Since III does not contain any of the given values, the only option that matches the sets and values correctly is option A) I.a II.b III.c, but III.c=4 is not in $P$. Therefore, none of the options perfectly match the sets and values given. **Final answer:** None of the options A, B, C, or D correctly match all sets with the given values.