Mathematical Reasoning

Mathematics is interesting is not a logical sentence. It may be interesting for some persons and may not be interesting for others.
this is not a proposition . holds.

Inverse of is
The contrapositive of is
contrapositive of is

Converse of is , then the contrapositive of is .

Consider the following statements
Two triangles are identical
Two triangles are similar
Clearly, the given statement in symbolic form is
Its contrapositive is given by

Now, : Two triangles are not identical Two triangles are not similar
If two triangles are not similar, then these are not identical

p
q
r












T
T
T
F
F
F
F
T
F
T
T


T
T
F
F
F
T
F
T
F
F
F


T
F
T
F
T
F
T
F
F
T
T


T
F
F
F
T
T
T
F
F
T
T


F
T
T
T
F
F
F
T
F
T
T

F
T
F
T
F
T
F
T
F
F
T


F
F
T
T
T
F
F
T
F
T
T


F
F
F
T
T
T
F
T
F
T
T






















T
T
T
T
F
T
F
F
T


F
T
F
F
F
F
F
F
F


F
T
T
F
T
F
F
T
T


F
T
T
F
T
F
F
T
F


T
T
T
T
T
T
T
T
T

F
T
T
T
T
T
T
T
F


T
T
T
T
F
T
F
F
F


T
T
T
T
F
T
F
F
F





(a)
is a contradiction from table.
(b)
is a tautology from table.
(c)

, represent logical equivalence from table.
(d)

, represent logical equivalence from table.

Truth table




p
q















F
F
T
T
T
T
T
T
T
T
T
F
F


F
T
T
F
T
T
F
F
F
T
T
F
F


T
F
F
T
F
F
T
T
F
F
F
T
T


T
T
F
F
T
T
T
T
F
F
T
F
F





Here, we observe that the truth value of column and are same.
Hence,

p
q











T
T
F
F
T
F
T
T
F


T
F
F
T
F
T
T
F
T


F
T
T
F
T
T
F
T
T


F
F
T
T
T
T
T
T
T




From the above table, the truth values of and are same.

Given proposition is

Inverse of E (i) is

(a) Mathematics is interesting, is a correct statement.
(b) is a prime number, is not a correct statement.
(c) is an irrational number, is a correct statement.
(d) The sun is a star, is a correct statement.

The correct answer is (B) : If two lines are not parallel then they intersect in the same plane.