Mathematical Logic
1.
The area of a trapezium is given by the formula :
A=
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
1
( B + b ) x H,
2
Where ‘A’ is the area in square units,
‘B’ is the base in units, ‘b’ is the other base in units
and
‘H’ is the height in units,
find b if A = 40, B = 6 H = 8.
Write the negation of each of the following statements :
i) x is not a real irrational number
ii) complex numbers are real numbers
iii) Mridul is not cruel and he is not strict.
Determine the truth value of each of the following statements :
i) 3 + 3 = 6 iff 2 + 2 = 4
ii) 3 + 3 = 7 iff 5 + 1 = 2
iii) If 5 < 2, then – 2 < - 5.
Construct the truth table for the following : ( p  q)  (~ p)
Construct the truth table for ( p  q)  (~ p  q)
Show that : i) ( p  q)  (~ p) is a tautology
ii) ( p  q)  (~ p) is a contradiction.
Show that
p  ( p  q) is a tautology.
Prove, by construction of truth tables, that p  q ~ q ~ p ,
where ‘~’ denotes ‘negation’ and ‘  ’ denotes if and only if.
Prove that ~ [(~ p)  q]  p  (~ q) .
Prove that ( p  q)  r and p  (q  r ) are not logically equivalent.
Determine whether the following are logically equivalent or not :
[~ ( p  q)  ( p  (~ r )]  [(~ p)  (~ q)] and ( p  r ) .
Write the duals of the following :
i) ( p  q)  t
ii)
( p  t)  r
Mathematical Logic
Formulae
TRUTH TABLES
1) Truth table for ~ p
table for p  q
P
T
F
2)
~P
F
T
p
T
T
F
F
4) Truth table for p  q
P
T
T
F
F
q
T
F
T
F
Truth table for p
pq
T
F
T
T

q
3) Truth
pq
T
F
F
F
q
T
F
T
F
p
T
T
F
F
q
T
F
T
F
p q
T
T
T
F
5) Truth Table p  q
P
T
T
F
F
q
T
F
T
F
pq
T
F
F
T
1) p v q = q v p ,
p q =q p
Commutative property
2) (p v q) v r = p v (q v r) , ( p  q )  r = (q  r)
Associative property
3) p v (q  r ) = (p v q)  (p v r) , p  (q v r ) = (p  q) v (p  r) Distributive property
4) ~ (p v q)  ~ p  ~ q , ~ (p  q)  ~ p v ~ q } Demorgan’s Laws
5) p  q  ~ p v q
} Equivalent statements
p  q  (p q)  (q p)  (~p v q) ( ~q v p) } Equivalent statements
6) p v (p  q) = p , p  (p v q ) = p
Absorption laws
7) If “ t” denotes the tautology and ‘c’ denotes the contradiction, then for any statement ‘p’ :
i) p v t = t ; p v c = p
ii) p  c = c
8) (i) p v ~ p =t (ii) p  ~ p = c (iii) ~ (~p) =p (iv) ~ t = c (v) ~c=t ( Complement Laws)
( Note:- All these laws are used to simplify the switching circuits.)
9) To write the converse, inverse and contra positive of a given statement :
Given statement : p  q
Converse : q p ( change the order of prime statements)
Inverse : ~ p q ( change the sign of each prime statement)
Contra positive :- ~ q~p( change the order as well as sign of each prime statement)
10) Switching circuit