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I'm unsure on how to answer this type of question, please can someone explain how to answer these step by step:

$1)$ Recall that the Cartesian product $A\times A$ is defined as the set $$\{(x,y):x\in A\land y\in A\}.\tag{I}$$ Thus if for example $A=\{1,2,3\}$, $$A\times A=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}.$$

Consider a set $A\neq \varnothing$ where the number $|A|$ of elements of $A$ is $20$ less than the number $|A\times A|$ of elements in $A\times A$. Thus $|A|+20=|A\times A|$.

Determine the number of elements in $A$. $$$$

$2)$ Recall that the Cartesian product $A \times A$ is defined as $(\rm I)$. Thus if for example $A = \{a, b, c\}$, $$A \times A = \{(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)\}.$$

Consider a set $A\neq \varnothing$ where the number $|A|$ of elements of $A$ is $30$ less than the number $|A \times A|$ of elements in $A \times A$. Thus $|A| + 30 = |A \times A|$.

Determine the number of elements in $A$.

Thank you in advance!

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closed as off-topic by Lord Shark the Unknown, Gregory J. Puleo, Mr Pie, blub, N. F. Taussig Apr 26 at 9:34

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    $\begingroup$ Here's a reference for MathJax you might find useful, which is preferred for writing up and rendering your math text on this site. $\endgroup$ – Eevee Trainer Apr 26 at 2:15
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Hint:

You should note that, for finite nonempty sets $S$,

$$|S\times S| =|S|^2$$

and in general for an $n$-ary product

$$\left| \prod_{i=1}^n S \right| = |S|^n$$

With this in mind, your equations become quadratics in terms of a variable that is the cardinality of $A$, which will give you the number of elements in $A$.


Hint $\#2$:

Also don't forget: if you end up with non-integer solutions, you either did it wrong, or the problem is poorly framed - "half an element" doesn't make sense. (You won't get non-integer solutions in this scenario, at least if you did the math right.)

Similarly, you can't have negative numbers of elements in a set.

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  • $\begingroup$ Thank you but what does it mean by where the number of elements of A is 20 less than AxA $\endgroup$ – AGRHQ Apr 26 at 2:26
  • $\begingroup$ It means that $$|A| = |A \times A| - 20$$ This is equivalent to what you wrote in the body of the post. $\endgroup$ – Eevee Trainer Apr 26 at 2:27
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    $\begingroup$ All you have to do is recall that $|A \times A| = |A|^2$. Then the equation I wrote in my previous comment becomes $$|A| = |A|^2 - 20$$ If it help makes it clearer, let $x = |A|$. Then you have that $$x = x^2 - 20$$ and your goal is to solve for $x$, just like in your high school algebra classes. $\endgroup$ – Eevee Trainer Apr 26 at 2:31
  • $\begingroup$ In a sense. You might want to be careful on how you write that: there is a big difference between the product of sets $A\times A$ and the cardinality of a set $|A|$. But yes, the cardinality of the product of finite sets is the product of their cardinalities. In any event, regarding how to "show it on paper," I would just convert the equations into quadratics (like I did earlier) and then solve those. $\endgroup$ – Eevee Trainer Apr 26 at 2:44
  • $\begingroup$ Yup, it would be correct $\endgroup$ – Eevee Trainer Apr 26 at 2:50
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**Hint: ** Try to think of this algebraically. The Cartesian product of $A$ TIMES $A$ is $A^2$. How can you set this up into an algebraic equation? It should be some sort of quadratic equation.

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