Solve the simultaneous equations and set theory problems: 1) x + 2y = 7 and 2x − 2y = 2. 2) P = primes < 15, Q = even numbers < 15. Find P U Q and if P, Q are disjoint.

$$(x + 2y) + (2x - 2y) = 7 + 2$$ $$3x = 9$$

$$x = \frac{9}{3}$$ $$x = 3$$

$$2y = 7 - 3$$ $$2y = 4$$ $$y = \frac{4}{2}$$ $$y = 2$$

The solution is $\boxed{x=3, y=2}$.

3b. Consider the sets:
$P = \{x : x \text{ is a prime number less than 15}\}$
$Q = \{x : x \text{ is an even number less than 15}\}$

i) List all the elements of sets P and Q:
Prime numbers less than 15 are numbers greater than 1 that are only divisible by 1 and themselves.

$$P = \{2, 3, 5, 7, 11, 13\}$$

Even numbers less than 15 are integers divisible by 2.

$$Q = \{2, 4, 6, 8, 10, 12, 14\}$$

ii) Find P U Q:
The union of P and Q includes all elements present in either set, without repetition.

$$P \cup Q = \{2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14\}$$

iii) State whether P and Q are disjoint sets:
Disjoint sets have no common elements (their intersection is an empty set).
The intersection of P and Q is $P \cap Q = \{2\}$, since 2 is in both sets.
Since $P \cap Q \neq \emptyset$, the sets P and Q are not disjoint.