Bab 3 - Mathematical reasoning

1. Determine whether each sentence below is a statement or not a statement. Tick a statement.

• A cuboid has 6 faces.

• 3x + 5 = 6

• (a + b)(a - b) = a² - b²

• Don't forget to bring workbook tomorrow.

• xy = 23

1. Determine whether the following statement are true or false.

'All fraction is smaller than one.'

• True

• False

Using the quantifier "all" or "some", write a true statement based on the following sentence.

' .................. polygons have diagonals.'

• All

• Some

Determine the antecedent and consequent of the following statement.

'If p < q, then q - p > 0.

• Antecedent : p< q

• Antecedent : q - p > 0

• Consequent : p < q

• Consequent : q - p > 0

Construct two implications based on the following sentence.

'M is divisible by 20 if and only if M is divisible by 2 and 10.'

Implication 1 :

Implication 2 :

Write down Premise 2 to complete the following argument.

Premise 1: If x is greater than zero, then x is a positive number.

Premise 2 :

Conclusion :8 is a positive number.

Make a general conclusion by induction for the sequence of numbers 11, 22, 43, .. which follows the following pattern.

11 = 5(2)¹ + 1

22 = 5(2)² + 2

43 = 5(2)³ + 3

• 5(2)ⁿ + n, n= 1,2,3,...

• 5n + n , n = 1, 2, 3

• 5(2n) + n, n = 1, 2, 3, ....

• 5n² + n, n = 1, 2, 3,...

What is the conclusion for the following argument.
Premise 1 : If  $ax^2\ +\ bx\ +\ c\ =\ 0$  has real roots, then  $b^2\ -\ 4ac\ \ge\ 0$  

Premise 2 :  $2x^2\ +\ px\ -\ 2\ =\ 0\$  has real roots.
Conclusion :

•  $p^2\ -\ 4\left(2\right)\left(-2\right)\ge0$  

•  $p^2\ -\ 4\left(2\right)\left(-2\right)\le0$  

•  $p^2\ -\ 4\left(2\right)\left(-2\right)\ =0$  

Determine whether the given arguments are strong or weak. Hence, determine whether the strong argument is cogent or not cogent.

Premise 1 : 1 is a prime number.

Premise 2 : 2 is a prime number.

Premise 3 : 3 is a prime number.

Premise 4 : 5 is a prime number.

Conclusion : All prime numbers are divisible by 1 and itself.

• This argument is strong and cogent.

• This argument is weak and not cogent.

• This argument is weak and cogent.

• This argument is strong and not cogent.

Are the following argument valid and sound?

Premise 1 : All multiples of 16 are even numbers.

Premise 2 : 64 is a multiple of 16.

Conclusion : 64 is an even number.

• Valid and sound.

• Valid but not sound.

• Not valid but sound.

• Not valid and not sound.