Solved Assignments and Study Material for Desklib

assignment - 1
discrete mathematics
MACQUARIE UNIVERSITY INTERNATIONAL COLLEGE
APRIL 10, 2019

Ques 1.
a) Use the laws of logic to simplify the expression: (p → ¬q) ∧ ¬(¬p → q) until it involves only a
single p and a single q.
Sol.
p q ¬ p ¬ q p → ¬q ¬p → q ¬(¬p →
q)
(p → ¬q)
∧ ¬(¬p →
q)
0 0 1 1 1 0 1 1
0 1 1 0 1 1 0 0
1 0 0 1 1 0 1 1
1 1 0 0 0 1 0 0
E=p' q' + p q'
E=q' ( p+ p' )
E=q '
b) ) Show using truth tables that p ↑ q ≡ p ⊕ (p → q) where ↑ is NAND and ⊕ is XOR.
Sol.
p q p ↑ q (p → q) p ⊕ (p → q)
0 0 1 1 1
0 1 1 1 1
1 0 1 0 1
1 1 0 1 0
From the above table it is clearly proved that p ↑ q ≡ p ⊕ (p → q)
Ques 2. Use a formal contrapositive proof to prove that if c ≥ ab, then either a ≤ √ c or b ≤ √ c.
Sol.
By using the concept of contrapositive proof
If c ≤ ab then a ≥ √b∨b ≥ √c
c ≤ ab
√c ≤ √a √b
For above equation ¿ be true ,
√c ≤ √b∨ √b≤ √a
If √ c ≤ √ b ,is obvious that √ c ≤b
¿ If √b ≤ √a ,is obvious that √ b≤ a

Hence proved
Ques 3. Write predicate logic statements to express the following English statements. Create and
explain suitable predicates as required; don’t write the predicate logic statements in English, use
logic symbols, variables and predicates, explaining any predicates you define. The domains of
interest are a class of students at a university and the things they do in their spare time.
a) There exists a student that plays tennis.
Sol.
P=Play ( student ,tennis )
∃ x , y : P( x , y)
b) All the students have a sport that they enjoy playing.
Sol.
P=Enjoy ( student , sport )
∀ x ,∃ y : P( x , y)
c) There are at least two students who do not enjoy photography.
Sol.
P=Enjoy ( student , photography )
∃ x ∃ y ∃ z (¬ ( x = y ) ¬ P(x , z )∧ ¬ P( y , z))
d) Every student who plays video games, also enjoys movies and reading.
Sol.
P=Play ( Student , Video game )
Q=Enjoy ( Student , Movie )
R=Enjoy ( Student , Reading )
∀ x ∃ y (P( x , y )→Q( x , y)∧ R( x , y))
Ques 4. For the point p= (−3
1 ) and the matrix A=(4 −1
2 2 ), find the following, showing all
working:
a) The result of a clockwise rotation on p by 90°.
Sol.
For clockwise rotation pint p must be multiplied by the matrix
m= ( cos θ sin θ
−sin θ cos θ )
θ=90 °

m= ( cos 90 ° sin 90°
−sin 90 ° cos 90 ° )
m= ( 0 1
−1 0)
mp= ( 0 1
−1 0 )(−3
1 )
mp= ( 0∗(−3 )+1∗1
−3∗(−1 ) +0∗1 )
mp= (1
3 )
p'=(1,3)
b) The multiplication Ap
Sol.
Ap= ( 4 −1
2 2 )(−3
1 )
Ap= (4∗(−3 )+ (−1 )∗1
2∗(−3 ) +2∗1 )
n=Ap=(−13
−4 )
c) Whether p, and the results of (a) and (b) above are co-linear?
Sol.
p= (−3
1 )
p'= (1
3 )
n=(−13
−4 )
p . p' =−3∗1+1∗3=0
|| p||= √ 10
|| p '||= √ 10=3.16
θ=cos−1 p . p '
||p||∨|p'
|∨¿ ¿