Discrete Mathematics
1.3.74
We know (1)a→b≡¬a∨b, (2)¬(a∧b)≡¬a∨¬b, (3)¬(a∨b)≡¬a∧¬b,
(4)a∨(¬b∧¬a) ≡a∨¬b, (5)a∨¬a≡a, (6)a∨T≡T
So (p∨q)∧(¬p∨r)→(q∨r)
≡¬((p∨q)∧(¬p∨r))∨(q∨r) (use(1))
≡(¬(p∨q)∨¬(¬p∨r))∨(q∨r) (use(2))
≡((¬p∧¬q)∨(p∧¬r))∨(q∨r) (use(3))
≡(¬p∧¬q)∨(p∧¬r)∨q∨r
≡(q∨(¬p∧¬q))∨(r∨(p∧¬r))
≡(q∨¬p)∨(r∨p) (use(4))
≡q∨r∨(p∨¬p)
1.4.84
(c)(i)∃x¬S(x),S(x):x can swim
(ii)∃x(C(x)∧¬S(x)),C(x):x is a person in your class
(d)(i)∀xQ(x),Q(x):x can solve quadratic equations
(ii)∀x(C(x)→Q(x))
(e)(i)∃x¬R(x),R(x):x wants to be rich
(ii)∃x(C(x)∧¬R(x))
