UPSKILL MATH PLUS

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### Theory:

Consider the set of all natural numbers.

Roster form of the set: $A\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\left\{1,\phantom{\rule{0.147em}{0ex}}2,\phantom{\rule{0.147em}{0ex}}3,\phantom{\rule{0.147em}{0ex}}4,\phantom{\rule{0.147em}{0ex}}5,...}\right\$.

Set builder form:$A\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\left\{x:x\phantom{\rule{0.147em}{0ex}}\mathrm{is}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathrm{natural}\phantom{\rule{0.147em}{0ex}}\mathrm{number}\right\}\right\$.

Thus, it can be concluded that $$1, 2, 3,...$$ are belongs to the set $$A$$.

But the element $$0$$ doesn't belongs to the set $$A$$.
Here comes the list of symbols we often use to denote the sets.
 Symbol Meaning Example $\in$ belongs to Suppose $A\phantom{\rule{0.147em}{0ex}}=\left\{3,4,5\right\}$ then $$3$$$\in$$$A$$. $\notin$ does not belongs to Suppose $A\phantom{\rule{0.147em}{0ex}}=\left\{3,4,5\right\}$ then $$6$$$\notin$$$A$$. $$:$$ or $$|$$ such that The set builder form $A\phantom{\rule{0.147em}{0ex}}=\left\{3,4,5\right\}$ is $A\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\left\{x:x\phantom{\rule{0.147em}{0ex}}\mathrm{is}\phantom{\rule{0.147em}{0ex}}a\phantom{\rule{0.147em}{0ex}}\mathrm{natural}\phantom{\rule{0.147em}{0ex}}\mathrm{number},\phantom{\rule{0.147em}{0ex}}3\le x\le 5}$ $\mathrm{ℕ}$ the set of all natural numbers $\mathrm{ℕ}=\left\{1,2,3,...\right\}$ $$W$$ the set of all whole numbers $$W = {0, 1, 2, 3,...}$$ $\mathrm{ℤ}\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}I$ the set of all integers $\mathrm{ℤ}\phantom{\rule{0.147em}{0ex}}=\left\{...,-3,-2,-1,\phantom{\rule{0.147em}{0ex}}0,1,\phantom{\rule{0.147em}{0ex}}2,3,...\right\}$ $\mathrm{ℤ}+$ the set of all positive integers $\mathrm{ℤ}+\phantom{\rule{0.147em}{0ex}}=\left\{1,\phantom{\rule{0.147em}{0ex}}2,3,4,...\right\}$ $\mathrm{ℚ}$ the set of rational numbers $\mathrm{ℚ}=\left\{\frac{p}{q},\phantom{\rule{0.147em}{0ex}}q\ne 0\phantom{\rule{0.147em}{0ex}}\mathrm{with}\phantom{\rule{0.147em}{0ex}}p\phantom{\rule{0.147em}{0ex}}\mathrm{and}\phantom{\rule{0.147em}{0ex}}q\phantom{\rule{0.147em}{0ex}}\mathrm{are}\phantom{\rule{0.147em}{0ex}}\mathrm{integers}}$ $\mathrm{ℚ}+$ the set of positive rational numbers $\mathrm{ℚ}+=\left\{\frac{p}{q},\phantom{\rule{0.147em}{0ex}}q\ne 0\phantom{\rule{0.147em}{0ex}}\mathrm{with}\phantom{\rule{0.147em}{0ex}}p\phantom{\rule{0.147em}{0ex}}\mathrm{and}\phantom{\rule{0.147em}{0ex}}q\phantom{\rule{0.147em}{0ex}}\mathrm{same}\phantom{\rule{0.147em}{0ex}}\mathrm{sign}\phantom{\rule{0.147em}{0ex}}\mathrm{integers}}$ $\mathrm{ℝ}$ the set of real numbers $\mathrm{ℝ}=\left\{x|-\mathrm{\infty }\le x\le \mathrm{\infty }\right\}$ $\mathrm{ℝ}+$ the set of positive real numbers $\mathrm{ℝ}+=\left\{x|\phantom{\rule{0.147em}{0ex}}0