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Ajad
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- Math
- Physics
- Trigonometry
- Geometry
What is mathematics? Mathematics is a game played according to certain simple rules with meaningless marks on paper.
- Math
- Physics
- Trigonometry
- Geometry
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About Ajad
I am done my graduation from csjmu Kanpur with physics & now preparing for B.Ed in Bundelkhand university
I have1.5 year experience of teaching
I have done work as a physics teacher
from class 9 to 12 both online & offline
contact ((concealed information))
place- orai
sub mathematics & physics
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All languages in which the lesson is available :
English
Mathematical Statements Revisited
Recall, that a ‘statement’ is a meaningful sentence which is not an order, or an
exclamation or a question. For example, ‘Which two teams are playing in the
PROOFS IN MATHEMATICS
2022-23
314 MATHEMATICS
Cricket World Cup Final?’ is a question, not a statement. ‘Go and finish your homework’
is an order, not a statement. ‘What a fantastic goal!’ is an exclamation, not a statement.
Remember, in general, statements can be one of the following:
• always true
• always false
• ambiguous
In Class IX, you have also studied that in mathematics, a statement is
acceptable only if it is either always true or always false. So, ambiguous sentences
are not considered as mathematical statements.
Let us review our understanding with a few examples.
Example 1 : State whether the following statements are always true, always false or
ambiguous. Justify your answers.
(i) The Sun orbits the Earth.
(ii) Vehicles have four wheels.
(iii) The speed of light is approximately 3 × 105
km/s.
(iv) A road to Kolkata will be closed from November to March.
(v) All humans are mortal.
Solution :
(i) This statement is always false, since astronomers have established that the Earth
orbits the Sun.
(ii) This statement is ambiguous, because we cannot decide if it is always true or
always false. This depends on what the vehicle is — vehicles can have 2, 3, 4, 6,
10, etc., wheels.
(iii) This statement is always true, as verified by physicists.
(iv) This statement is ambiguous, because it is not clear which road is being referred
to.
(v) This statement is always true, since every human being has to die some time.
Example 2 : State whether the following statements are true or false, and justify your
answers.
(i) All equilateral triangles are isosceles.
(ii) Some isosceles triangles are equilateral.
(iii) All isosceles triangles are equilateral.
(iv) Some rational numbers are integers.
2022-23
PROOFS IN MATHEMATICS 315
(v) Some rational numbers are not integers.
(vi) Not all integers are rational.
(vii) Between any two rational numbers there is no rational number.
Solution :
(i) This statement is true, because equilateral triangles have equal sides, and therefore
are isosceles.
(ii) This statement is true, because those isosceles triangles whose base angles are
60° are equilateral.
(iii) This statement is false. Give a counter-example for it.
(iv) This statement is true, since rational numbers of the form , p
q
where p is an
integer and q = 1, are integers (for example,
3 3
1 = ).
(v) This statement is true, because rational numbers of the form , p
q
p, q are integers
and q does not divide p, are not integers (for example,
3
2 ).
(vi) This statement is the same as saying ‘there is an integer which is not a rational
number’. This is false, because all integers are rational numbers.
(vii) This statement is false. As you know, between any two rational numbers r and s
lies 2
r s +
, which is a rational number.
Example 3 : If x < 4, which of the following statements are true? Justify your answers.
(i) 2x > 8 (ii) 2x < 6 (iii) 2x < 8
Solution :
(i) This statement is false, because, for example, x = 3 < 4 does not satisfy 2x > 8.
(ii) This statement is false, because, for example, x = 3.5 < 4 does not satisfy 2x < 6.
(iii) This statement is true, because it is the same as x < 4.
Example 4 : Restate the following statements with appropriate conditions, so that
they become true statements:
(i) If the diagonals of a quadrilateral are equal, then it is a rectangle.
(ii) A line joining two points on two sides of a triangle is parallel to the third side.
(iii) p is irrational for all positive integers p.
(iv) All quadratic equations have two real roots.
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