Question 1
(i) p= `` It is raining.'', q = ``I have an umbrella.''
(p Ù(Øq)) Ú(Øp)
(ii) p=``It is raining.'', q=``I can stay at home.''
(Øp) ® (Øq)
Question 2
(i)
| p | q | Øq | (Ø1) Ùp | p ® ((Øq) Ùp) |
| T | T | F | F | F |
| T | F | T | T | T |
| F | T | F | F | T |
| F | F | T | F | T |
(ii)
| p | q | p® q | (p ® q) Ùq | [(p ® q) Ùq] ® p |
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | F | T |
Question 3
p=`` The two lines given are parallel.''
q=`` The two lines given have the same slope.''
Statement is p® q.
Converse (q ® p)
If the two lines given have the same slope, then they are parallel.
Inverse (Øp ® Øq)
If the two lines given are not parallel then they do not have the same
slope.
Contrapositive ((Øq) ® (Øp))
If the two lines given do not have the same slope then they are not
parallel.
Negation (Ø(p® q) º Ø(Øp Úq) º p Ù(Øq))
The two lines given are parallel and do not have the same slope.
Question 4
p=``Sandra finishes her work'',
q=``It snows'',
r=``Sandra goes to the basket ball game.''
|
| p | q | r | Øq | (Øq ® r) | p® (Øq ® r) | [p® (Øq ® r)] Ù(Øq) |
| T | T | T | F | T | T | F |
| T | T | F | F | T | T | F |
| T | F | T | T | T | T | T |
| T | F | F | T | F | F | F |
| F | T | T | F | T | T | F |
| F | T | F | F | T | T | F |
| F | F | T | T | T | T | T |
| F | F | F | T | F | T | T |
| p ® r | { p® (Øq ® r) Ù(Øq)} ® (p ® r) |
| T | T |
| F | T |
| T | T |
| F | T |
| T | T |
| T | T |
| T | T |
| T | T |
The following table shows this is a tautology, therefore the argument is valid.
Question 5
(i) For every item in the whole world if it is a cat then it is not a
dog.
(ii) There exists some item in the whole world that is not a dog and not
a cat.
Question 6
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Question 7
(i) False
(ii) False
(iii) True
(iv) False
(v) True
Question 8
(i) A ÈB = {1,2,3,5,6,{1,2}}
(ii) A ÇC = { 1,2,8}
(iii) B Ç[`C] = Æ
(iv) (A Ç[`B]) ÈC = E
(v) [`(([`B] È[`C]))] = {3,5,8}
Question 9
7 students played no games.