1,277
edits
Changes
all truth tables inserted...
5.
(a)
{|
! p || q || r || p.-r || p.q || p.q ⊃ r || p.-r ∨ (p.q ⊃ r)
|F || F || F || F || F || T || T
|}
(b)
{|
! p || q || r || s || -p.-q || q.-s || p.-r.s || -p.-q ∨ q.-s ∨ p.-r.s
|-
| T || T || T || T || F || F || F || F
|-
| T || T || T || F || F || T || F || T
|-
| T || T || F || T || F || F || T || T
|-
| T || T || F || F || F || T || F || T
|-
| T || F || T || T || F || F || F || F
|-
| T || F || T || F || F || F || F || F
|-
| T || F || F || T || F || F || T || T
|-
| T || F || F || F || F || F || F || F
|-
| F || T || T || T || F || F || F || F
|-
| F || T || T || F || F || T || F || T
|-
| F || T || F || T || F || F || F || F
|-
| F || T || F || F || F || T || F || T
|-
| F || F || T || T || T || F || F || T
|-
| F || F || T || F || T || F || F || T
|-
| F || F || F || T || T || F || F || T
|-
| F || F || F || F || T || F || F || T
|}
(c)
{|
! p || q || r || (p∨q)≡-r || p∨(-q ⊃ r) || (p∨q)≡-r . p∨(-q ⊃ r)
|-
|T || T || T || F || T || F
|-
|T || T || F || T || T || T
|-
|T || F || T || F || T || F
|-
|T || F || F || T || T || T
|-
|F || T || T || F || T || F
|-
|F || T || F || T || T || T
|-
|F || F || T || T || T || T
|-
|F || F || F || F || F || F
|}
6.
(a)
{|
! p || p || p$p
|-
|T || T || F
|-
|F || F || F
|}
(b)
{|
! q || p || q$p
|-
|T || T ||
|-
|T || F ||
|-
|F || T ||
|-
|F || F ||
|}
(c)
{|
! p || q || p$q || (p$q)$p
|-
|T || T ||
|-
|T || F ||
|-
|F || T ||
|-
|F || F ||
|}
(d)
{|
! p || q || r || (p$q)$r
|-
| T || T || T ||
|-
| T || T || F ||
|-
| T || F || T ||
|-
| T || F || F ||
|-
| F || T || T ||
|-
| F || T || F ||
|-
| F || F || T ||
|-
| F || F || F ||
|}
(e)
{|
! p || q || r || (p⊃r) || (p⊃r)$q
|-
| T || T || T ||
|-
| T || T || F ||
|-
| T || F || T ||
|-
| T || F || F ||
|-
| F || T || T ||
|-
| F || T || F ||
|-
| F || F || T ||
|-
| F || F || F ||
|}
7. (a)
{|
! p || p || r || (p⊃r) || (p⊃p) . (p⊃r)
|-
| T || T || T ||
|-
| T || T || F ||
|-
| F || F || T ||
|-
| F || F || F ||
|}
(b)
{|
! p || r || q || (r⊃q) || (p⊃q) || (r⊃q) . (p⊃q)
|-
| T || T || T ||
|-
| T || T || F ||
|-
| T || F || T ||
|-
| T || F || F ||
|-
| F || T || T ||
|-
| F || T || F ||
|-
| F || F || T ||
|-
| F || F || F ||
|}
(c)
{|
! p || r || q || (p⊃f(p,q,p)) || (q⊃f(p,q,p)) || (p⊃f(p,q,p)) . (q⊃f(p,q,p))
|-
| T || T || T ||
|-
| T || T || F ||
|-
| T || F || T ||
|-
| T || F || F ||
|-
| F || T || T ||
|-
| F || T || F ||
|-
| F || F || T ||
|-
| F || F || F ||
|}
(d)
{|
! p || r || q || ((p⊃q)⊃r) || ((q⊃p)⊃r) || ((p⊃q)⊃r) . ((q⊃p)⊃r)
|-
| T || T || T ||
|-
| T || T || F ||
|-
| T || F || T ||
|-
| T || F || F ||
|-
| F || T || T ||
|-
| F || T || F ||
|-
| F || F || T ||
|-
| F || F || F ||
|}
8.
(a) Yes, as the truth-values for s#-t and -s#t are the same.
(b)
{|
! s || t || u || (s#t) || (s#t)#u || t#u || s#(t#u)
|-
| T || T || T ||
|-
| T || T || F ||
|-
| T || F || T ||
|-
| T || F || F ||
|-
| F || T || T ||
|-
| F || T || F ||
|-
| F || F || T ||
|-
| F || F || F ||
|}
9. (a) Compare the values of the 2nd and 3rd rows; if they are equivalent, then # is commutative. Otherwise, it isn't.
(b) Say P and Q are equivalent, i.e. both true. Then, for a connective to be anti-commutative, that means that P#Q is equivalent to -(Q#P). However, this is impossible as P#Q and Q#P are equivalent, so the negation of the latter cannot be equivalent to the former.
10.
(a) No. Say p = "God exists", but there is actually no God. Many people believe God exists is true. Say p = "oceans are blue", which they are. Many people believe oceans are blue is true. Thus the truth value of the statement depends on more than just the truth-value of its single component.
(b) No. Say p = "the sun's gravitational field exerts an attractive force on all objects", q = "some frogs are green"; then the whole statement is false. However, say p remains the same and q = "the planets orbit around the sun"; then, the whole statement is true. Thus the whole statement can be both true and false if p & q are both true, and it is not truth-functional as the truth-values of p & q alone do not determine its truth value.
(c) Yes; this is simply the "not" connective, whose output depends solely on the truth value of the input.