Question

Let A = {1, 2, 3} and B = {2, 3, 4, 5}. Find the cardinalities of the
following sets:
(i) A ∪ B
(ii) A ∩ B
(iii) A \ B
(iv) B \ A
(v) P(A ∪ B)

Exercise 1.2. Let A = {◦, {◦}, {∅}} and let B = {∅, {◦}}. Find the cardinalities of
the following sets:
(i) A ∪ B
(ii) A ∩ B
(iii) A \ B
(iv) A × B
(v) P(A)
Exercise 1.3. Find the cardinality of the set
\
n∈N
[0, 1/n].

Exercise 1.4. Prove distributive property (1.3.6), that for any sets A, B, and C, we
have that

(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C).

Answer 1

$Exercise~1.1\\(i)~A\cup B=\{1,2,3,4,5\}\\ Cardinality~of~A\cup B=5\\ (ii)~A\cap B=\{2,3\}\\ Cardinality~of~A\cap B=2\\ (iii)~A-B=\{1\}\\ Cardinality~of~A-B=1\\ (iv)~B-A=\{4,5\}\\ Cardinality~of~B-A=2\\ (v)~P(A\cup B)=power~set~of~A\cup B=2^5=32\\$$Exercise~1.2\\(i)~Cardinality~of~A\cup B=A=3\\ (ii)~A\cap B=B\\ Cardinality~of~A\cap B=B=2\\ (iii)~A-B=\{o\}\\ Cardinality~of~A-B=1\\ (iv)~Cardinality~of~A\times B=A~Cartesian~product~ B=2\times 3=6\\ (v)~Cardinality~of~P(A)=2^3=8\\$

$Exercise~1.3\\ \left\{ n\in \mathbb{N}:~[0,1/n]\right\}=\left\{ 0,1,1/2,...,1/n\right \}.\\ Hence~its~cardinality=n+1.\\ Exercise~1.4\\ x\in (A\cap B)\cup C\Rightarrow x\in (A\cap B)~or~x\in C\\ \Rightarrow (x\in~A~and~x\in B)~or~x\in C\\ \Rightarrow (x\in A~or~x\in C)~and~(x\in B~or~x\in C)\\ \Rightarrow x\in(A\cup C)\cap (B\cup C)\\ Therefore,~(A\cap B)\cup C\subseteq (A\cup C)\cap (B\cup C).......(1)$

$x\in (A\cup C)\cap (B\cup C)\Rightarrow x\in (A\cup C)~and~x\in (B\cup C)\\ \Rightarrow (x\in A ~or~x\in C)~and~(x\in B~or~x\in C)\\ \Rightarrow (x\in A~and~x\in B)~or~x\in C\\ \Rightarrow x\in (A\cap B)\cup C.........(2)\\ From~(1)~and~(2),~we~get\\ (A\cap B)\cup C=(A\cup C)\cap (B\cup C)$