If we have these three sets: \begin{align*} A &= \{3\}\\ B &= \{3,4,5\}\\ C &= \{3,4,5\} \end{align*}
And the main statement (which I am supposed to prove or disprove) is: $$A \cup C = B \cup C. $$
- $A \cup C = \{3\} \cup \{3,4,5\}$
- $B \cup C = \{3,4,5\} \cup \{3,4,5\}$
(First of all, I am wondering if statement 1 and 2 are true?)
Is it true that $A \cup C = B \cup C?$
If it is false, does it have to be
$A \cup C = \{3,4,5\} \cup \{3,4,5\}$ and $B\cup C= \{3,4,5\} \cup \{3,4,5\}?$ I don't understand if it is enough if one of the sets on one side, is equal to one of the sets on the other side - or if both sides have to be equal to each other with both sets.
I hope this was understandable, and will be super happy if anyone can help :))))
An element in a union of sets is an element in at least one from those sets.
Two sets are equal when they contain exactly the same elements.
So, don't stop at substitution, rather, perform the union operation and then compare the results
$\begin{split}A\cup C &= \{3\}\cup\{3,4,5\} &=\{x:x{\in}A\vee x{\in}C\} \\ & = \{3,4,5\} \\[2ex] B\cup C & = \{3,4,5\}\cup\{3,4,5\}\\ & = \{3,4,5\}\\[2ex]\therefore\quad A\cup C&=B\cup C \end{split}$