Fundamentals of Mathematic

Question 1
(a) Let A, B, C and D be the following statements:
A: Andrew comes for the exam.
B: Bryan comes for the exam.
C: Charles comes for the exam.
D: Daniel comes for the exam.
Write the following compound sentences using logical connectives:
(i) If Daniel comes for the exam, then if Charles doesn’t come then Andrew come
for the exam.
(ii) Charles comes for the exam provided that Daniel doesn’t come, but if Daniel
comes, then Bryan doesn’t come.
(5 marks)
(b) Let , and r be the following statements:
f: The brick is falling
a: The brick is above me
r: I run
(i) Write the following argument using , and r and logical connectives:
If the brick is falling and above me, then I will run.
The brick is falling.
The brick is not above me.
Therefore, I will not run.
(ii) Employ a truth table to test the validity of the above argument.
(11 marks)
MTH105 Tutor-Marked Assignment
SINGAPORE UNIVERSITY OF SOCIAL SCIENCES (SUSS) Page 3 of 5
(c) Construct a chain of logical connectives to show that (p ∧ q) ∧ [(q ∧ ¬r) ∨ (p ∧ r)] is
logically equivalent to ¬(p → ¬q).
Do not use truth tables here and give a reason for each line.
(8 marks)
Question 2
(a) Consider the following statement:
There is at least a student in SUSS who has not visited the National Library or visited
Changi Airport.
Let S(x), N(x) and C(x) be the predicates “x is an SUSS student” and “x has visited
the National Library” and “x has visited Changi Airport” respectively.
(i) Rewrite the statements using the predicates S(x), N(x) and C(x) together with
the universal and/or existential quantifier.
(ii) Write the negation of the statement using the predicates S(x), N(x) and C(x)
together with the universal and/or existential quantifier.
(6 marks)
(b) Use the rules of inference to show the validity of the following argument. Give a
reason for each line of your proof.
x, (F(x)  G(x)) x, (F(x)  H(x)) x, (H(x)  I(x)) x, (G(x)  I(x))
(9 marks)
Question 3
(a) (i) State the contrapositive of the following statement:
Let . If
3 2 3 2
y  yx  x  xy
, then
y  x .
(ii) Use the method of proof by contrapositive to prove the above statement.
(7 marks)
(b) Use a suitable method to prove that there are no rational solution to the equation
1 0
3
x  x  
or give a counter-example to show otherwise.