Find the elements in the intersection of square numbers and multiples of 3

Key takeaway: Set questions like this are really about reading the definitions carefully. First list the elements of each set inside the universal set, then intersect them.

The universal set is

$U=\{41,42,43,44,45,46,47,48,49\}.$

Now identify each set:

• $P$ = square numbers in $U$ = $\{\, 49 \,\}$
• $R$ = multiples of 3 in $U$ = $\{42,45,48\}$

The intersection $P\cap R$ means elements that are in both sets. Since $49$ is not a multiple of 3,

$\boxed{P\cap R=\varnothing}.$

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Pitfalls the pros know 👇 It is easy to forget that the universal set limits everything here. Even if a number is a square or a multiple of 3 in general, it only counts if it lies between 40 and 50. Also, $\varnothing$ is the correct answer when there are no common elements; do not write $0$ or leave it blank.

What if the problem changes? If the question had asked for $P\cup R$, then you would list all square numbers or multiples of 3 in the universal set, without repeating any element. If the universal set were expanded to include $36$, then $P\cap R$ would no longer be empty because $36$ is both a square number and a multiple of 3.

Tags: intersection, universal set, square number