Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$,$A = \{1, 2, 5\}$,and $B = \{6, 7\}$. Find $A \cap B'$.

Fill in the blank to make the following a true statement:
$U^{\prime} \cap A = \ldots$

Taking the set of natural numbers as the universal set,write down the complement of the following set:
$A = \{ x : x \in N \text{ and } 2x + 1 > 10 \}$

$A$ and $B$ are events such that $P(A)=0.42$,$P(B)=0.48$ and $P(A \cap B)=0.16$. Determine $P(\text{not } A)$.

If $U = \{a, b, c, d, e, f, g, h\}$,find the complement of the set $B = \{d, e, f, g\}$.

For each positive real number $\lambda$,let $A_\lambda$ be the set of all natural numbers $n$ such that $|\sin(\sqrt{n+1}) - \sin(\sqrt{n})| < \lambda$. Let $A_\lambda^c$ be the complement of $A_\lambda$ in the set of all natural numbers. Then,