問71
頻出度★★☆
次のベン図の黒色で塗りつぶした部分の検索条件はどれか。
平成21年度 春期 ITパスポート試験 問71
テクノロジ系基礎理論 › 離散数学
This question's explanation and your answers so far.
解説
問 71
- ア
正解 The correct answer is ア. The black area in the figure is the lower half of the tall lens-shaped region formed where circle C on the left overlaps circle B on the right — the part left over after circle A takes away the upper portion. Check this area against each of the three circles to see whether it is inside or outside: first, it is part of the overlap of C and B, so it belongs to both B and C. On the other hand, its upper edge is cut off by the arc of circle A, so it lies outside circle A. In other words, this area satisfies the three conditions “not A”, “B”, and “C” at the same time, so joining them with and gives (not A) and B and C as the answer. In the tall lens where B and C overlap, the upper half is the triple overlap that also includes A, but that part is not shaded black. This fact is clear evidence that not A is required.
Incorrectイ
(not A) and (B or C) represents the area that is “outside A” and at the same time “inside at least one of B or C”. This would also take in the parts that belong to B alone and to C alone (the parts that do not overlap A), producing a wide area close to the size of two whole circles. The black part in the figure is only the place that is inside both B and C, so this connector must be and, not or.
Incorrectウ
Because (not A) or (B and C) is joined by or, anything that is merely “outside A” already satisfies the condition. That takes in even the background outside the diagram, which no circle covers at all, and on top of that the region where A, B, and C all overlap is added in from the (B and C) side. This is not a formula that represents only the narrow shaded part of the figure.
Incorrectエ
(not A) or (B or C) means “outside A” or “inside B” or “inside C”, and the only thing this formula does not cover is the part that belongs to A alone. It therefore points to the widest possible area — nearly everything, including the background outside the diagram — which is the exact opposite of the narrow black part.
ここがポイント
- The procedure for rewriting a Venn diagram as a logical formula can be purely mechanical: for the shaded area, check each circle to see whether the area is inside or outside it; write the condition as it is when inside, and add not when outside; since these conditions hold at the same time, join them with and. Here the area is outside A, inside B, and inside C, which gives (not A) and B and C.
- and takes the intersection, so the more conditions you add the narrower the area becomes, while or takes the union, so the more conditions you add the wider it becomes. In particular, joining not A with or takes in even the background outside the circles, so when the formula has to represent a small shaded area, the options that join not A with or (ウ and エ) can be ruled out first.
用語