TIC1201 AY2019/20 Sem 2 - Assignment 1: Logic and Problem Solving

This document presents a complete solution for Assignment 1 of the TIC1201 course, focusing on logic and reasoning. The solution addresses three main questions. The first question involves proving a logical statement using truth tables and implication rules. The second question tackles a syllogism problem, analyzing statements about actors, musicians, and singers to derive valid conclusions using logical deduction. The third question provides a series of true/false answers. Finally, the solution includes a hat color puzzle, explaining the reasoning behind determining the color of a hat based on given clues and deductive reasoning. The solutions are presented with detailed explanations and justifications to aid understanding.

Q1
((i can) → ((can do) → (it it))) → can ≡ can∧ ∼ ∧ ∨ ∼
Let
i = A
can = B
do = C
it = D
To prove:
((A B) → ((B C) → (D D))) → B ≡ B∧ ∼ ∧ ∨ ∼
We know that (D D) is always True.∨ ∼
So LHS reduces to :
((A B) → ((B C) → True)) → B∧ ∼ ∧
According to the Implication Truth Table:
A B A →B
False False True
False True True
True False False
True True True
We can notice that in Implication logic, if any statement implies True predicate, the result is also
always True.
Hence our LHS can be reduced to :
((A B) → True) → B∧ ∼
Similarly :
True → B
Again from the Truth table of implication logic, we can notice that True →'X' results in 'X'.
Hence LHS can further be reduced to :
True → B = B
Hence LHS = RHS
Hence proved the given statement.
Q2.
Statements:
1. All actors are musicians.
2. No musician is a singer.
3. Some singers are dancers.
Conclusions:

1. Some actors are singers.
2. Some dancers are actors.
3. No actor is a singer.
(a)
For statement 2:
x(~Musician(x) → Singer(x))∀
For statement 3:
∃x(Singer(x) → Dancer(x))
(b)
Only conclusion (3) ,i.e., No actor is a singer is valid if all the above three statements are considered to
be True. Statement 1 states that all the actors are musicians and statement 2 states that none of the
musician is a singer. Hence, it can be easily concluded from the two statements that None of the actor is
a singer which makes only Conclusion 3 as valid.
Q3.
(a)
False
(b)
True
(c)
False
(d)
True
(e)
True
(f)
True
(g)
True
Q4.
The color of A's Hat is Black.
Explanation:
C says that he does not know the color of his hat. Also, he can see A's hat as well as B's hat. From C's
statement, we can conclude that Red & white combination is not worn by A or B. At least one Black hat
is worn either by A or by B, else C would be sure that he has a Black Hat.
Now, B again confirms that he also do not know the color of his own hat, although he can see A's hat.

This means that :
suppose if A's hat was red, B would be confirmed that his own hat is black because red & white
combination is already eliminated.
Similarly, if A's hat was white, again B would be sure of his own black hat because red would be
eliminated.
Hence, we can definitely conclude that A's hat is black.