Maths

PROOFS IN MATHEMATICS 333
EXERCISE A1.6
1. Suppose a + b = c + d, and a < c. Use proof by contradiction to show b d.
2. Let r be a rational number and x be an irrational number. Use proof by contradiction to
show that r + x is an irrational number.
3. Use proof by contradiction to prove that if for an integer a, a 2 is even, then so is a.
[Hint : Assume a is not even, that is, it is of the form 2n + 1, for some integer n, and then
proceed.]
4. Use proof by contradiction to prove that if for an integer a, a 2 is divisible by 3, then a is
divisible by 3.
5. Use proof by contradiction to show that there is no value of n for which 6 n ends with the
digit zero.
6. Prove by contradiction that two distinct lines in a plane cannot intersect in more than
one point.
A1.8 Summary
In this Appendix, you have studied the following points :
1. Different ingredients of a proof and other related concepts learnt in Class IX.
2. The negation of a statement.
3. The converse of a statement.
4. Proof by contradiction.