The following is a question from "A First Course in Probably $9^{th}$ Edition" Question 4 from the Section 2 Self Study.
Let $A$ denote the event that the midtown temperature in Los Angeles is $70^◦F$, and let $B$ denote the event that the midtown temperature in New York is $70^◦F$. Also, let $C$ denote the event that the maximum of the midtown temperatures in New York and in Los Angeles is $70^◦F$. If $P(A) = .3$,$P(B) = .4$, and $P(C) = .2$, find the probability that the minimum of the two midtown temperatures is $70^◦F$.
The solution did the following:
Let $D$ be the even that the minimum temperature is $70^◦F$. Then $$ \begin{align} P(A \cup B) = P(A) + P(B) - P(A\cap B) \\ P(C \cup D) = P(C) + P(D) - P(C\cap D) \end{align} $$
Then we fill in what we know: $$ \begin{align} P(A \cup B) = 0.3 + 0.4 - P(A\cap B) \\ P(C \cup D) = 0.2 + P(D) - P(C\cap D) \end{align} $$
And then the author throws something out there that I cannot justify:
$$ \text{Since }P(A \cup B) = P(C \cup D) \\ \text{and } P(A \cap B) = P(C \cap D) $$
How do we know that these two are equal? I have not the slightest idea of why this would be true. Is it because $P( A \cup B) = 1$?
Consider what $A, B, C, D$ actually mean. $A \cup B$ is the event "A or B", i.e. that at least one of the temperatures is $70^◦F$. $C \cup D$ is the event that either the maximum temperature or the minimum temperature is $70^◦F$. These two statements have exactly the same meaning. Same for $A \cap B$ and $C \cap D$; they both denote the event that both temperatures are $70^◦F$.