# Cartesian Product and Sets - Discrete Mathematics [closed]

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 $$A$$.

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

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

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.

• Thank you but what does it mean by where the number of elements of A is 20 less than AxA – AGRHQ Apr 26 at 2:26
• It means that $$|A| = |A \times A| - 20$$ This is equivalent to what you wrote in the body of the post. – Eevee Trainer Apr 26 at 2:27
• 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. – Eevee Trainer Apr 26 at 2:31
• 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. – Eevee Trainer Apr 26 at 2:44
• Yup, it would be correct – Eevee Trainer Apr 26 at 2:50

**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.