Given set P = 2, 4, 6. Which of the following is a subset of P? A. 2, 2, 4 B. 2, 3, 4 C. 2, 4, 6

$$n(E \cup H) = n(U) - n(E \cup H)'$$ $$n(E \cup H) = 45 - 3 = 42$$

$$n(E \cup H) = n(E) + n(H) - n(E \cap H)$$ $$42 = 25 + 21 - n(E \cap H)$$ $$42 = 46 - n(E \cap H)$$ $$n(E \cap H) = 46 - 42 = 4$$

$$n(\text{Economics only}) = n(E) - n(E \cap H)$$ $$n(\text{Economics only}) = 25 - 4 = 21$$

The correct option is C.
The number of students who take Economics only is $\boxed{\text{C. } 21}$.

:
Given set $P = \{2, 4, 6\}$.
A set $A$ is a subset of $P$ if every element in $A$ is also an element in $P$.

• A. $\{2, 2, 4\}$ is equivalent to $\{2, 4\}$. Both 2 and 4 are in $P$. So, $\{2, 2, 4\}$ is a subset of $P$.
• B. $\{2, 3, 4\}$. The element 3 is in this set, but 3 is not an element of $P$. Therefore, $\{2, 3, 4\}$ is not a subset of $P$.
• C. $\{2, 4, 6\}$. All elements (2, 4, 6) are in $P$. So, $\{2, 4, 6\}$ is a subset of $P$.
• D. $\emptyset$. The empty set is a subset of every set. So, $\emptyset$ is a subset of $P$. The correct option is B. The set that is not a subset of $P$ is $\boxed{\text{B. } \{2, 3, 4\}}$.

:
We need to find the equivalent expression for $(M' \cap N)'$.
Using De Morgan's Laws, which state that $(A \cap B)' = A' \cup B'$ and $(A \cup B)' = A' \cap B'$.
Also, $(A')' = A$.

Apply De Morgan's Law to $(M' \cap N)'$:

$$(M' \cap N)' = (M')' \cup N'$$ $$(M' \cap N)' = M \cup N'$$

The correct option is C.
$(M' \cap N)'$ is equivalent to $\boxed{\text{C. } (M \cup N')}$.

:
Let $S$ be the set of teachers who teach pure Science and $T$ be the set of teachers who teach Social Science.
Given:
Total teachers $n(U) = 65$
Teachers teaching pure Science $n(S) = 45$
Teachers teaching Social Science $n(T) = 32$
Teachers who do not teach any subject $n(S \cup T)' = 7$

$$n(S \cup T) = n(U) - n(S \cup T)'$$ $$n(S \cup T) = 65 - 7 = 58$$

$$n(S \cup T) = n(S) + n(T) - n(S \cap T)$$ $$58 = 45 + 32 - n(S \cap T)$$ $$58 = 77 - n(S \cap T)$$ $$n(S \cap T) = 77 - 58 = 19$$

The correct option is D.
The number of teachers who teach both subjects is $\boxed{\text{D. } 19}$.