Find the number of elements in P U QP U Q, if set PP comprises all multiples of 44 less than 200200 and Set QQ comprises all odd multiples of 33 less than 100100.
Question
Find the number of elements in P U QP U Q, if set PP comprises all multiples of 44 less than 200200 and Set QQ comprises all odd multiples of 33 less than 100100.
Solution
To solve this problem, we need to find the number of elements in each set and then combine them.
Step 1: Find the number of elements in set P Set P comprises all multiples of 44 less than 200. To find the number of elements, we divide 200 by 44, which gives us approximately 4.54. Since we can't have a fraction of an element, we round down to 4. So, there are 4 elements in set P.
Step 2: Find the number of elements in set Q Set Q comprises all odd multiples of 33 less than 100. To find the number of elements, we first find the number of multiples of 33 less than 100, which is 100 divided by 33, or approximately 3.03. We round down to get 3. However, not all of these are odd, so we need to check each one. The multiples of 33 less than 100 are 33, 66, and 99. Of these, only 33 and 99 are odd, so there are 2 elements in set Q.
Step 3: Combine the sets The union of sets P and Q, denoted P U Q, is the set of all elements that are in P, or Q, or both. Since there are no common elements between P and Q, we simply add the number of elements in each set to get the total number of elements in P U Q. So, there are 4 + 2 = 6 elements in P U Q.
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