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

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Lord Shark the Unknown, Gregory J. Puleo, Mr Pie, blub, N. F. Taussig
If this question can be reworded to fit the rules in the help center, please edit the question.

• 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.