Exercise - 2

1. Using the truth tables find whether the following are tautologies or contradictions or neither.

Algebra of statements

1. Idempotent laws:
(a) p v p equivalence p
Laws of Algebra of statements.











(b)p ^ p equivalence p











2.Associative laws:








3. Commutative laws:









4. Distributive laws:







5. De Morgan's laws:

Tautologies and Contradictions

Tautologies and Contradictions:
Some compound statements contain only T in the last column of their truth tables. Such statements are called tautologies.

A compound statement which contains only F in the last column of its truth table is called contradiction.

Two compound statements are said to be logically equivalent if their truth tables are identical. We denote the logical equivalence of two compound statement P and Q by P=Q.

Example1:
The statement "p or not p" i.e., p v (~p)is a tautology.
This can be seen by the truth table.










Example2:
The Statement "p and notp" i.e., p^(~p) is a contradiction

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Statements and Sets Fill in the blanks

Exercise - 1















2. Write the conjunction and disjunction of the given statements.
(i) 5 is an odd number; 5 is positive
5 is an odd number or 5 is positive.
5 is an odd number and 5 is positive.
Similarly we can write it for (ii),(iii),(iv) and (v)

3. Write each of the following statements using the appropriate connectives ^,V,=>,<=>,~. And state their truth values.

(i)5+7=10 and 4+3=7
5+7=10 ^ 4+3=7
F ^ T= F

(ii) 7 is odd or 7 is prime
7 is odd v 7 is prime
T V T = T

(iii) If x+2=0 then x=-2
If then means implication
x+2=0 => x=-2
T => T = T

(iv) x+2=0 iff x=-2
Biimplication
x+2=0 <=> x=-2
T <=> T = T

(V) 7 not equal to 10
~(7=10)- T

4. Rewrite the following statements using the connective symbols. Here p, q are any two statements.

(i) notp
~p
(ii) p or q
p V q

(iii) p and q
p ^ q

(iv) p only if q
p => q

(v)p iff q
p <=> q

(vi) p and not q
p^(~q)

(vii) (~p) or q
(~p)Vq

(viii)(either p) or (not p)
(either p)V(~p)

(ix) p or not p
pV(~p)

(x)(not p) and (not q)
(~p)^(~q)





















6. State the truth values of the following disjunctions
(i) 4+3=7 or 5x4=21
T ^ F = T
1 + 0 = 1
(ii) 4 divided by 2=1 or 2 divided by 1 = 4
F ^ F = F
0 + 0 =0
(iii) 3 divided by 2=1 or 3x2=6
F^T=T
0 + 1 = 1
(iv) 4x5=20 or 4 divided by 2 = 2
T^T=T
1 + 1 = 1

7.State the truth values of the following conjunctions.
(i) 20 divided by 10=2 and 20x10=200
T v T = T
1 x 1 =1
(ii) 10+2=12 and 10 divided by 2 =5
T v T = T
1 x 1 =1
(iii) 10+15 = 20 and 15-10=5
F v T = F
0 x 1 = 0
(iv) 15x3=45 and 15+3=20
F v F = F
0 x 0 = 0

8.State the truth values of the following implications.
(i)If 3+2=5 then 1x0=0
T => T = T
(ii) If 3x6=20 then 2+7=9
F => T = T
(iii) If 5x7=30 then 2 divided by 1=2
F => T = T
(iv) If 6x7=42 then 6 divided by 2=4
T => F= F

9.State which of the following are true statements
(i) 4+3=10 <=> 4x3=12
F <=> T = F
(ii) 4x7=20 <=> 4 divided by 7=1
F <=> F = T
(iii) 5x8=40 <=> 8-2=5
T <=> F = F
(iv)6-3=3 <=> 6x3=18
T <=> T = T

10. Write the converse, inverse and contrapositive of the following conditions.
(i) If two triangles are congruent then they are similar.
Converse: If two triangles are similar then they are congruent.
Inverse: If two triangles are not congruent then they are not similar.
Contrapositive: If two triangles are not similar then they are not congruent.

(ii) If in a triangle ABC, AB=AC then AngleB=AngleC
Converse: If in a triangle ABC, AngleB=AngleC then AB=AC.
Inverse:If in a triangle ABC, AB not equal to AC then AngleB not equal to AngleC.
Contrapositive: If in a triangle ABC, AngleB not equal to AngleC then AB not equal to AC.
Similarly we can do the rest.

11. Let p be "She is beautiful" and q be "She is happy". Write each of the following Statements that are in symbolic form into english sentences.
(i)p ^ q
She is beautiful and she is happy.
Similarly we can do the rest.

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