Assignment 3
(Concepts covered: Biconditional statements, contrapositive, converse,
quantifiers, negation of quantifiers, some proofs involving these concepts)
(1) Consider the statement: ‘A polygon is a triangle if and only if it has exactly three sides’. Explain what is ‘necessary’ and ‘sufficient’ condition
in this.
(2) Let x be a real no. Prove by the direct method that the real no. x
equals 2 if and only if x3 − 2x2 + x = 2.
(3) Check the following with the truth tables.
(a) P =⇒ Q is equivalent to it’s contrapositive −Q =⇒ −P
(b) P =⇒ Q is not equivalent to it’s converse Q =⇒ P
(4) Write the converse and contrapositive for the following :
(a) Suppose f : R → R is a function defined on an interval containing
a real no. α. If f is differentiable at x = α then f is continuous at
x = α.
(b) Suppose that a and b are integers. If a · b = 0 then either a = 0 or
b = 0.
(5) Check with the truth table that
[(P =⇒ Q) ∧ (Q =⇒ R)] =⇒ (P =⇒ R)
(6) Explain the negation of the following statements.
(a) ∀ real numbers x, x3 ≥ x2 .
(b) ∃ a real no. x such that x + 1 = 0.
(c) ∃ a real no. y such that ∀ real no.s x, x + y = 0
(d) ∀ real numbers x, P (x) =⇒ Q(x)
(7) Give an example to show that the following statement is false.
‘For all real no.s x and y, if x 6= y then xy + xy > 2’.
Instead now consider the statement, ‘For all real no.s x and y, if
x 6= y, x > 0 and y > 0 then
x y
+ >2
y x
What assumptions need to be made for a proof by contradiction? Complete the proof. ( Hint: Consider the negation of this statement)
(8) Consider the statement : ‘If r is a real number such that r2 = 20 then,
r is irrational. Carefully write down all the conditions that you would
assume if you are setting up for proof by contradiction. Complete a
proof by contradiction for this statement. √
(9) ‘For all positive real numbers a and b , if ab 6= a+b
2 then a 6= b’. Is
this statement true or false? Prove the statement if it is true and give
a counter example if it is false. ( Hint: You can also think about the
contrapositive of this statment)
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