Let\ U = \left\{ 1,​2, 3,​ ...,2400 ​\right\}Let S be the subset of the numbers in U that are multiples of 3​, and let T be the subset of U that are multiples of 7.Find​ n(T) using a method similar to the one that showed that n(S)=800.

Isa Trevino 2021-08-01 Answered

Let U \(= \left\{ 1,​2, 3,​ ...,2400 ​\right\}\)
Let S be the subset of the numbers in U that are multiples of 3​, and let T be the subset of U that are multiples of 7.
Since \(\displaystyle{2400}\div{3}={800}\)​, it follows that \(n(S)=n(\left\{3 \cdot 1, 3 \cdot 2, \cdots, 3 \cdot 800\right\})=800\).
​(a) Find​ n(T) using a method similar to the one that showed that \(\displaystyle{n}{\left({S}\right)}={800}\).
(b) Find \(\displaystyle{n}{\left({S}\cap{T}\right)}\).
(c) Label the number of elements in each region of a​ two-loop Venn diagram with the universe U and subsets S and T.
Questions:Find n(T) ? Find n(SnT)

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Expert Answer

Arham Warner
Answered 2021-08-02 Author has 24173 answers

Step 1
Solution: \(\displaystyle{n}{\left(\cup\right)}={2400}\)
\(\cup = \left\{1,2,3, .....,2400\right\}\)
\(\displaystyle{2400}\div{3}={800}\)
\(n(s) = n \left\{3 \cdot 1, 3 \cdot 2, .... 3 \cdot 800\right\} = 800\)
\(\displaystyle{n}{\left({t}\right)}={n}{\left({7}\cdot{1},{7}\cdot{2},{7}\cdot{3},\ldots.,{7}\cdot{342}\right)}={342}\)
2394
Step 2
\(\displaystyle{n}{\left({t}\right)}={342}\)
\(\displaystyle{s}\cap{t}\) Multiple of 7 and \(\displaystyle{3}={21}\)
\(\displaystyle{n}{\left({s}\cap{t}\right)}={n}{\left({21}\cdot{1},{21}\cdot{2},\ldots..{21}\cdot{114}\right)}={114}\)
2394
\(\displaystyle{n}{\left({s}\cap{t}\right)}={114}\)

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