Video: AQA GCSE Mathematics Higher Tier Pack 3 • Paper 3 • Question 18

(a) A Venn diagram is shown below. Which of the following represents the shaded region on the Venn diagram? [A] A ∪ B′ [B] A′ ∩ B [C] A [D] B′. (b) A different Venn diagram is shown below. Which of the following represents the shaded region on the Venn diagram? [A] (A ∪ B)′ [B] A′ ∩ B′ [C] A ∩ B [D] (A ∩ B)′

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Video Transcript

Part a) A Venn diagram is shown below. Which of the following represents the shaded region on the Venn diagram? Is it A ∪ B dash, A dash ∩ B, A, or B dash.

Before answering this specific question, it is important to remember some of the notation for probability. Firstly, A dash means not A. This means that the fourth option B dash means the area that is not in B. The u or union symbol in the first option means “or.” This means that the first option represents the area that is in A or not in B.

The n or intersection symbol means “and” as seen in the second option. This demonstrates the area that is not in A and in B. Whilst you might have spotted already which one is the correct answer, let’s look at all four options and what their Venn diagrams would look like.

Let’s firstly consider the third option A. This consists of everything inside the circle A. Clearly, this answer is incorrect. Secondly, let’s look at B dash. This is the area of everything not in B. This is the area of everything that is not in circle B. Once again, this is clearly a wrong answer.

The second option not A and B needs to include the area that is not in A, but also in B. It must be outside of circle A, but inside of circle B. This includes everything inside circle B except the intersection or overlap between the two circles. It is clear from this that this is the correct answer.

Whilst we have found the correct answer, it is worth just confirming what A ∪ not B would look like. This is any area that is in A or not in B. Firstly, we could circle the area of A. We also need to shade any area left that is not in B. This leaves us with an area similar to but not exactly the same as the fourth option. Clearly, this is also incorrect.

The shaded region on the Venn diagram given is represented by not A ∩ B.

A different Venn diagram is shown below. Which of the following represents the shaded region on the Venn diagram? Is it not A ∪ B, is it not A ∩ not B, is it A ∩ B, or is it not A ∩ B?

Once again, it is worth sketching the four different options. The third option A ∩ B is the overlap of circles A and circles B. Clearly, this is not the correct answer. However, you might notice that the correct answer is the exact opposite of this. Everything else is shaded except the intersection. This means that the correct answer is not A ∩ B.

This is shown in the fourth option as it is everything that is not in the intersection. This means that the shaded region on the Venn diagram is represented by not A ∩ B. Once again, we will shade the Venn diagrams for the other two options for completeness.

The second option not A ∩ not B involves shading everything that is not in circle A and not in circle B, everything outside the two circles. Once again, this is clearly not correct. The union of A and B is everything inside circle A or inside circle B, everything contained within both circles. This means that not A ∪ B would be everything outside the two circles.

This means that the notation in option one and option two would give the same shaded region, both of which are incorrect. In this part of the question, the fourth option was correct.

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