The compound statement with "and " is 
25 is a multiple of 5 and 8.
Since p is true and q is false, so the compound statement with "and" is not true.
Thus the statement" p and q" is not valid.
Now the compound statement with "or" is
25 is a multiple of 5 or 8. Since p is true and q is false, so the compound statement with "or" is true Thus the statement "p or q" is valid.

(i) You watch television if and only if your mind is free.
(ii) You get an A grade if and only if you do all the homework regularly.
(iii) A quadrilateral is equiangular if and only if it is a rectangle.

(i) If you log on to the server, then you have a password.
(ii) If it rains, then there is traffic jam.
(iii) If you can access the website, then you pay a subscription fee

(i) ~ p: There exists a positive real number x such that x-I is not positive.
(ii) ~ q: There exists a cat which does not scratch.
(iii) ~ r: There exists a real number x such that neither x > 1 nor x < l.
(iv) ~ s: There does not exist a number x such that 0 < x < l.

(i) Component statements of the compound statement are
p: 100 is a multiple of 2
q: 100 is a multiple of 5
Now both p and q are true. Thus the given compound statement is valid.
(ii) The component statements of the compound statement are
p: Square of an integer is positive.
q: Square of an integer is negative.
Let us assume that p is false
\(\therefore \) x²< 0 for x \(\in \) Z
q is true
So when p is false, q is true.
Thus the given compound statement is valid.

(i)Here p: You were born in India.
q: You are a citizen of India.
Now ~ p: You were not born in India.
~ q: You are not a citizen of India.
The contrapositive of the given statement is:
If you are not a citizen of India then you were not born in India.
The converse of the given statement is:
If you are a citizen of India then you were born in India.
(ii)Here p: A triangle is equilateral.
q: A triangle is isosceles.
Now ~ p : A triangle is not equilateral.
\(\sim \)q: A triangle is not isosceles
The contrapositive of the given statement is:
If a triangle is not isosceles then it is not equilateral.
The converse of the given statement is:
If a triangle is isosceles then it is equilateral.
(iii)Here p: x is an even number.
q: x is divisible by 4.
Now ~ p: x is not an even number.
\(\sim \)q: x is not divisible by 4.
The contrapositive of the given statement is:
If x is not divisible by 4 then x is not an even number.
The converse of the given statement is:
If x is divisible by 4 then x is an even number

(i) If you earn a living then you work.
(ii) If a number is even then it is multiple of 2.
(iii) If you wish to pass in examination you must work hard.

(i)This statement makes use of exclusive 'or'. Since a geometrical figure can not be a quadrilateral and also a pentagon.
(ii) This statement makes use of inclusive 'or'. Since a geometrical figure can be a polygon as well as a parallelogram.

(i)The component statement has the connecting word 'and'. Component statements are
p: The sun is cold.
q: the moon rotates around the earth.
(ii)The component statement has the connecting word 'or'. Component statements are
p: The school works.
q: A holiday is declared.
(iii)The component statement has the connecting word 'or'. Component statements are.
p: Ankita is rich.
q: Ankita is happy

(i)The component statements are
p: The sun shines
q: it rains Both the component statements p and q are true.
(ii) The component statements are
p: India is a democracy
q: India is a monarchy
The component statement p is true whereas the component statement q is false.

(i) Negation of statement is: There exists a natural number which is not greater than 0.
(ii) Negation of statement is: The sum of 4 and 6 is not equal to 10.
(iii) Negation of statement is: A whole number is not an integer

The conjunction of the statement p and q is given by 
\(p\wedge q\) : It is cold and it is raining
The disjunction of the statement p and q is given by
\(p\vee q\) : It is cold and it is raining

The component statements of the given statement are
p : A natural number is odd
q : Square of a natural number is odd.
The given statement is 'if p then q'.
It can be expressed in five different ways as follows
(i) p\(\Rightarrow \)q i.e. x is an odd natural number \(\Rightarrow \) x² is an odd natural number.
(ii) P is a sufficient condition for q i.e. knowing that a natural number is odd is sufficient to conclude that its square is odd.
(iii) p only if q i.e. a natural number is odd only if its square is odd.
(iv) q is necessary condition for p i.e. when a natural number is odd, its square is necessarily odd.
(v) \(\sim \)q\(\Rightarrow \sim \)p i.e. if the square of a natural number is not odd, then the natural number is not odd.

The conditional statement of given statement is 'If you take dinner, then you will get a sweet dish.'

(i) T, because statement 'Delhi is in India' is true
(ii) F, because statement 'Chennai is in pakistan' is false
(iii) T, becuse statement '5 + 6 = 11' is true