Learning Task 3
A. Prove the following using Direct Proof
1. Given: Odd and even integers
Prove: the sum of odd and even integers is odd integer
Solution:
Supply the missing statement and reason
Answers
Explanation:
1. Reason:
2. Definition of odd integer
3. Definition of even integer
5. sum of integers
2. mn = (2k + 1)(2k)
= 4k2 + 2k product of integers
= 2(2k2 + k) factor out 2
= 2r let r = 2k2 + k
mn = 2r by definition of even integer
3. let n = 2k
3n + 5 = 3(2k) + 5 by substitution
= 6k + 5 multiply
= 6k + 4 + 1 renaming 5
= 2(3k + 2) + 1 factor out 2
= 2m + 1 let m = 3k + 2
3n + 5 = 2m + 1 is odd number by
definition.
B.
1. Given x = 3
Prove : 3x + 5 ≠ 10
Assume that 3x + 5 = 10
3x = 5
x = 5/3
This shows that the assumption
is not true. Hence the conclu-
sion 3x + 5 ≠ 10 is true.
2. Given: Triangle ABC is an
isosceles triangle.
Prove: Base angle cannot be
920.
Assume that the base angle is
920
Remember that the base angles
of isosceles triangle are equal
and the sum of the measure of
the angles of the triangle is 1800
A + B + C = 1800
A + 92 + 92 = 1800
This is not possible since the
two angles adds up to more
than 1800 already. Therefore
the assumption is false. The
conclusion that cannot be 920 is
true.
4. Given: x = 5
Prove: 2x + 4 ≠ 12
Assume that 2x + 4 = 12
2(5) + 14 =12
10 + 14 = 12 is false
Hence the conclusion 2x + 4 ≠ 12
is true.
Answer:
That's true
Explanation:
every even integer is in the form of 2q (q belong to any natural number)