Question Video: Representing Three given Sets Using Venn Diagram | Nagwa Question Video: Representing Three given Sets Using Venn Diagram | Nagwa

Question Video: Representing Three given Sets Using Venn Diagram Mathematics

If 𝑋 = {6, 0, 3, 7, 8}, π‘Œ = {8, 3, 5, 2}, and 𝑍 = {8, 0, 1, 4, 5}, which Venn diagram represents the sets? [A] Diagram A [B] Diagram B [C] Diagram C [D] Diagram D

03:55

Video Transcript

If 𝑋 equals the set of numbers six, zero, three, seven, eight; π‘Œ is equal to the set of numbers eight, three, five, two; and 𝑍 equals eight, zero, one, four, five, which Venn diagram represents the sets?

A starting point in this question might be to consider set 𝑋 and see which of our diagrams contain the numbers six, zero, three, seven, and eight. Unfortunately, this alone does not help as all four of our options have these five numbers. Set π‘Œ contains the numbers eight, three, five, and two. Once again, all four circles contain these four numbers. We know that set 𝑍 contains the numbers eight, zero, one, four, and five. Options (A), (C), and (D) all contain these five numbers in circle 𝑍. This means that we can only eliminate option (B) using this start point.

In questions such as this, it is far more important to look at the intersections or overlaps between the circles. The intersection notation is written as shown. Let’s first consider those numbers that appear in set 𝑋, set π‘Œ, and set 𝑍. The only number that appears in all three sets is eight. This means that we must have an eight in the intersection of all three circles. Once again, options (A), (C), and (D) all satisfy this. Let’s now consider which numbers appear in set 𝑋 and π‘Œ, 𝑋 and 𝑍, and π‘Œ and 𝑍. The only number apart from eight that appears in set 𝑋 and set π‘Œ is three. This means that the set of numbers in the intersection 𝑋 and π‘Œ is eight and three. The number three must appear in the overlap between set 𝑋 and set π‘Œ.

In options (C) and (D), this is true. However, in set (A) there is no number in this intersection, so we can rule out this option. The number zero exists in set 𝑋 and in set 𝑍. This means that the intersection of 𝑋 and 𝑍 contains eight and zero. We must have a zero in the overlap between 𝑋 and 𝑍. This is true for option (C), but not for option (D) as there is no number in the intersection between 𝑋 and 𝑍 only. This suggests that option (C) is the correct answer.

We will check this by considering the intersection of π‘Œ and 𝑍. The number five appears in set π‘Œ and set 𝑍. It also appears on the Venn diagram in the intersection of these two circles. The intersection of set π‘Œ and set 𝑍 needs to contain eight and five, which is correct. The Venn diagram that represents the three sets given is option (C). In this type of question, the most important part is to look at the intersection of the different sets.

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