Thursday, June 11, 2009
Exercise - 2
1. Using the truth tables find whether the following are tautologies or contradictions or neither.
Tuesday, June 9, 2009
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:
(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
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
Financial help for poor students
Hi
If you have come across any bright students coming from poor financial
background, who have finished their 10^th standard this year (April
2009) and scored more than 80%, please ask them to contact the NGO -
Prerana (supported by Infy foundation). The NGO is conducting a written
test on 14^th June and those who clear the test will be eligible for
financial help for their further studies.
Please ask the students to contact the people mentioned below to get the
form :Sarasawati Mob no. *9900906338
*#580, Shubhakar, 44^th cross, 1^st ‘A’ main road, Jayanagar 7^th block,
Bangalore
Mr. Shivkumar(9986630301) - he sits at Hanumanthnagar office
Ms. Bindu(9964534667) - she sits at Yeshwantpur office
If you have come across any bright students coming from poor financial
background, who have finished their 10^th standard this year (April
2009) and scored more than 80%, please ask them to contact the NGO -
Prerana (supported by Infy foundation). The NGO is conducting a written
test on 14^th June and those who clear the test will be eligible for
financial help for their further studies.
Please ask the students to contact the people mentioned below to get the
form :Sarasawati Mob no. *9900906338
*#580, Shubhakar, 44^th cross, 1^st ‘A’ main road, Jayanagar 7^th block,
Bangalore
Mr. Shivkumar(9986630301) - he sits at Hanumanthnagar office
Ms. Bindu(9964534667) - she sits at Yeshwantpur office
Sunday, June 7, 2009
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.
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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Polytechnic entrance is conducted by State Board of Technical Education.
Eligibility Criteria:
Candidates should have passed, SSC or it's equivalent examination.
Eligibility Criteria:
Candidates should have passed, SSC or it's equivalent examination.
Saturday, June 6, 2009
Engineering Colleges of Andhrapradesh
To view the information regarding different Engineering Colleges of Andhrapradesh.
Click Here
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Polytechnic Colleges of Andhrapradesh
To view the information regarding different polytechnic colleges of Andhrapradesh.
Andhra Region
Osmania Region
S.V University Region
Andhra Region
Osmania Region
S.V University Region
Converse, inverse and contrapositive of a conditional
Converse, inverse and contrapositive of a conditional:
Suppose p, q are two statements.
(i) 'If q then p' is called the converse of 'If p then q'.
(ii) 'If ~p then ~q' is called the inverse of 'If p then q'.
(iii) 'If ~q then ~p' is called the contrapositive of 'If p then q'.
Symbolically,
q=>p is the converse of p=>q.
~p=>~q is the inverse of p=>q.
~q=>~p is the contrapositive of p=>q.
Suppose p, q are two statements.
(i) 'If q then p' is called the converse of 'If p then q'.
(ii) 'If ~p then ~q' is called the inverse of 'If p then q'.
(iii) 'If ~q then ~p' is called the contrapositive of 'If p then q'.
Symbolically,
q=>p is the converse of p=>q.
~p=>~q is the inverse of p=>q.
~q=>~p is the contrapositive of p=>q.
Biconditional (or Bi implication)
Biconditional (or Bi implication):
The biconditional connective <=> (read as IF AND ONLY IF) is defined by the following truth table.
Note: If p and q have the same truth value, the biconditional p <=> q is true; if p and q have opposite truth values then
p <=> q is false.
Friday, June 5, 2009
Conditional (or Implication)
Conditional (or Implication) :
The conditional connective => (read as ONLY IF) can be defined by the following truth table.
Note that the compound statement p=>q is true always except the case when p is true and q is false.
Note: A true statement can't imply a false statement.
Example: State the truth values of the following implications.
(i) If 4 * 5=20 then 4+5=9 is of truth value T.
Because p: 4*5=20, q: 4+5=9 are true statements.
Conjunction
Conjunction:
The conjunction connective ^ (read as AND) is defined by the following truth table.
AND means multiplication.
1 * 1 = 1
1 * 0 = 0
0 * 1 = 0
0 * 0 = 0
Note : The conjunction of two statements is true only when each statement is true.
Disjuction
Disjuction:
The disjunction connective v (read as OR) is defined by the following truth table.
Note: The disjunction of two statements is false only when both the statements are false.
OR means addition.
1 + 1 = 1
1 + 0= 1
0 + 1 = 0
0 +0 = 0
Negation
Negation:
Given a statement p, another statement which is the negation of p can be formed by writing the word not before p and read as not p.
The negation connective (~) is defined by the following truth table.
Example 1: p:3 is a prime number.
~p: 3 is not a prime number.
Given a statement p, another statement which is the negation of p can be formed by writing the word not before p and read as not p.
The negation connective (~) is defined by the following truth table.
Example 1: p:3 is a prime number.
~p: 3 is not a prime number.
STATEMENTS AND SETS
In mathematics we use sentences that can be shown to be either true or false but not both. Such sentences are called Statements. Thus,
A statement is a sentence which is either true or false but not both.
Example 1: Suns rises in the east.
This is a statement which is true.
Example 2: 4+4=5
This is a statement which is false.
Example 3: The garden is very beautiful.This is only a sentence not a statement because we need further information to decide whether the sentence is true or false. We need information like "What is the name of the garden and where it is located?".
Connectives
Many Statements are the combination of simple statements with some connecting words. These words are called connectives. The statements that are composed of other simple statements with the connectives are called Compound Statements.
Observe the following examples.
Example1: p: All primes but two are odd.
q: 2 is an even prime.
We can form a number of compound statements composed of these two. A few are listed below.
(i) All primes but 2 are odd and 2 is an even prime.
i.e., p and q.
(ii) All primes but 2 are odd or 2 is an even prime.
i.e., p or q.
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A statement is a sentence which is either true or false but not both.
Example 1: Suns rises in the east.
This is a statement which is true.
Example 2: 4+4=5
This is a statement which is false.
Example 3: The garden is very beautiful.This is only a sentence not a statement because we need further information to decide whether the sentence is true or false. We need information like "What is the name of the garden and where it is located?".
Connectives
Many Statements are the combination of simple statements with some connecting words. These words are called connectives. The statements that are composed of other simple statements with the connectives are called Compound Statements.
Observe the following examples.
Example1: p: All primes but two are odd.
q: 2 is an even prime.
We can form a number of compound statements composed of these two. A few are listed below.
(i) All primes but 2 are odd and 2 is an even prime.
i.e., p and q.
(ii) All primes but 2 are odd or 2 is an even prime.
i.e., p or q.
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