I just started a course of discrete mathematics. We’ve seen basic sets and operations (union, intersection, etc…) with easy examples. But, I now have an homework in which I have to proceed with unions and intersections where the sets contains functions. I'm not so sure of what to do, I'm not asking to make this question, but if someone could help me by providing an example, it would be very nice. Thank you.
As an example,
$$
\begin{align}
A &= \{2x+3\ |\ x \in \{1,2,3,4\}\} \\
B &= \{2x + 1\ |\ x \in \mathbb{N}\} \\
C &= \{4x + 1\ |\ x \in \mathbb{N}\}
\end{align}
$$
Give a definition of the union A and B and it's cardinality
Give a definition of the intersection between A and B and it's cardinality
Best Answer
I think the easiest way is to write it out:
$$ \begin{array}{} A &=& \{2x+3\ |\ x \in \{1,2,3,4\}\} &=& \{5, 7, 9, 11\}\\ B &=& \{2x + 1\ |\ x \in \mathbb{N}\} &=& \{3,5,7,9,11,13,15,\dots\}\\ C &=& \{4x + 1\ |\ x \in \mathbb{N}\} &=& \{5,9,13,17,21,\dots\} \end{array} $$
Now you can apply union and intersection similar to easy examples.
If it's not clear I'll expand the answer.
(note: There are two common definitions of $\mathbb{N}$: one where $0 \in \mathbb{N}$ and one where $0 \not\in \mathbb{N}$. I assumed the latter.)