# 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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Arham Warner

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