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Given that π is a set containing the elements seven, 11, and 22, π is a set containing the elements one, 11, 21, and 22, and π and π are subsets of some universal set π, which of the following could be true?
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π is a set containing the element 22, π is a set containing the elements one, 11, and 21, π is a set containing the elements one, seven, 11, and 21, or π is a set containing the elements one, seven, 11, 21, and 22.
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This problem tests whether weβre able to say what a possible universal set might contain if we know some of its subsets.
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The question tells us that the universal set that weβre talking about is π, and π and π are the subsets.
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Letβs represent this as a Venn diagram.
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Letβs use this green square to represent the universal set π.
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This orange circle can represent set π.
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Notice that weβve drawn the circle completely inside the square.
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This is because π is a subset of the universal set π.
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Everything in π is inside π.
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And the numbers in π are seven, 11, and 22.
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And we can also draw a pink circle to represent set π, which is also a subset of the universal set.
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Now, when we look at the elements of set π, we can see that two of the numbers are also in set π.
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So, when we draw our pink circle, weβre going to have to make sure it overlapped with the orange one.
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So, the two numbers that both sets share are 11 and 22.
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And then, as weβve said already, set π contains the number seven too, and set π contains the numbers one and 21.
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So, this Venn diagram represents the question.
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Now, weβre given four statements about the universal set, π.
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And weβre asked which of the following could be true.
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Each statement gives the letter π and then tells us what the elements of that universal set are.
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So, the correct answer is the one that contains all the elements.
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When we look inside our green square, we can see the numbers one, seven, 11, 21, and 22.
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This means that the correct answer is the one that says π equals a set containing the elements one, seven, 11, 21, and 22, the five numbers that we can see inside the set.
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Notice the word could in the last question.
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Which of the following could be true?
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The reason why it could be true, and is not definitely true, is that the question doesnβt tell us that π and π are the only things within the universal set π.
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If they were the only sets within π, then this answer is correct.
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But of course, if there was another set there, then we would need to include the numbers in that too.
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Universal sets contain everything within them.
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So, given that π is a set containing the elements seven, 11, and 22, and π is a set containing the elements one, 11, 21, and 22.
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And given that both π and π are subsets of a universal set, which weβll call π.
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The statement that could be true is that π is a set containing the elements one, seven, 11, 21, and 22.