Logical Reasoning (Part 1: 3.1 Statements)

Hello and welcome! ;)

What do you think we will be learning today?

Poll

Can you guess what we will be learning today?

• financial management

• probability

• logical reasoning

• measures of dispersion

• how to make pizza for lunch later

Logical Reasoning, guys!

Ok so today we will revise back on this chapter starting off with

'Statements'.

Multiple Choice

Can you remember what a statement is?

• A statement is a sentence whether it is true or false

• A sentence can be a question, an instruction and an exclamation.

Multiple Choice

OK, let's try again...

Which sentence is a statement?

• What is the meaning of factor?

• Go and brush your teeth.

• Good morning!

• The highest common factor of 21 and 28 is 7.

Multiple Select

Ok, one more time...

Choose which of the following are statements...

• An equilateral triangle ABC has interior angle of

  $60\degree$ .

• Divide 18 with 3.

• $2+4=2\times4$

• $x>5$

Determining the Truth Value of a statement...

• Take note about quantifiers "all" and "some"...

• 'All' means that every object or case satisfies a certain condition. 'All' can also be substituted with words like 'every' or 'each'.

• 'Some' means a few and not every object or case satisfies a certain condition.'Some' can also be substituted with words like 'part of', 'a few of' or 'one of'

Multiple Select

Determine which statements are true.

• $p\left(p+q\right)=p^2+pq$

• $0.000123=1.23\times10^{-5}$

• $\left(-1\right)^{23}+\left(-1\right)^{24}=-1$

• $13^2-12^2=5^2$

Multiple Select

Determine which statements are false.

• All multiples of 8 are multiples of 4.

• All factors of 15 are factors of 10.

• Some rhombuses have four equal sides.

• Some prime numbers are even numbers.

Negating a Statement

• a process which denies a statement using the word 'not' or 'no'.

• A true statement can be changed to a false statement and vice versa using the word 'not' or 'no'.

• The symbol ~ (tilde) can be used to represent a negation.

Multiple Select

Which of these statements, after negating them would give you a false statement?

• {a, b} is a subset of {a, b, c}

• 3 is a factor of 3 and 5.

• 1.0 has two significant figures

• All straight lines have positive gradient.

Determining the truth value of a compound statement

• can be formed by combining two statements using the word 'and' or 'or'.

Multiple Select

From the following compound statements, which one is true?

• 16 is a multiple of 2 and 4

• $10^2-4^2=9^2\ and\ \ \ 5^2-4^2=3^2$

• $\frac{3}{7}>\frac{2}{7}\ and\ -\frac{4}{7}>-\frac{1}{7}$

• $10\%\ can\ be\ written\ as\ \frac{1}{10}\ and\ 0.1$

Multiple Select

Determine which of the following compound statements are false.

• 81 is a perfect square or a pentagon has five sides.

• $-m^{-1}=m\ or\ the\ solution\ of\ inequality\ -2x>6\ is\ x>-3$

Antecedent and Consequent of an Implication

• for an implication 'if p, then q', p is the antecedent and q is the consequent.

• Example, 

  Antecedent:

   $A\subset B$  
  Consequent:  $A\cap B\ =\ A$  
  If  $A\subset B$  , then  $A\cap B=A$  
  
  

• 'if p then q' can be written as  $p\ \Longrightarrow\ q$  

Implication 'if and only if'

• For the statement 'p if and only if q', two implications can be writtern as:

• Implication 1: If p, then q

• Implication 2: If q, then p

• 'p if and only if q ' can be written as

   $p\Longleftrightarrow q$  

Fill in the Blanks

Which are the two implications based on the following statement.

5x = -20 if and only if x = -4.

Fill in the Blanks

Construct a statement in the form 'if and only if' from each of the following pairs of implications.
Implication 1: If x = -2 then x - 6 = -8

Implication 2: If  x - 6 = -8, then x = -2.

Converse, Inverse and Contrapositive of an Implication

• Converse of an Implication: 'if p, then q' is 'if q, then p'.

• The converse of an implication may NOT have the same truth value as the implication.

• Example: State the converse of each of the following implications and hence, determine whether the converse is true or false.

• If x > 5, then x > 4. Converse: If x > 4, then x > 5 (The converse is false)

• If x > y, then y - x > 0. Converse: If y - x > 0, then x > y ( The converse is true)

Fill in the Blanks

State the converse of the implication and hence, determine whether the converse is true or false.

If x > 0, then x > -2.

Inverse of an Implication

• Implication: if p, then q

• Inverse: If not p, then not q

• The inverse of an implication may NOT have the same truth value as the implication

• Example: State the inverse of the following implication and hence, determine whether the inverse is true or false.

• If y - x > 0, then y > x.

• Implication: If y - x > 0, then y > x.

• Inverse: If y - x < 0, then y < x. (True)

Fill in the Blanks

State the inverse of the implication and determine whether the inverse is true or false.

If n < - 2, then n < - 1.

Fill in the Blanks

State the inverse of the implication and hence determine whether the inverse is true or false.

If m < 0, then m < 3.

Contrapositive of an implication

• Implication: if p, then q

• Contrapositive: if not q, then not p

• Contrapositive of an implication is always true if the implication is true.

• Example: Implication:   $if\ \theta\ =\ 90\degree\ ,\ then\ \sin\ \theta\ =\ 1$  

• Contrapositive:  $if\ \theta\ne90\degree,\ then\ \sin\theta\ne1$  (True)

Fill in the Blanks

State the contrapositive of the following implication and determine whether the contrapositive is true or false.

If x is an even number , then x is divisible by 2.

Determine the counter-example (denial) to negate the truth of a particular statement.

• involves quantifier (All/Some), compound statement (and/or) , negation (not) and implication (if p, then q)

• Example: All prime numbers have two factors only. (Not all prime numbers have two factors only.)

• An hour is equal to sixty minutes and 1 km is equal to 1000 m. ( An hour is not equal to sixty minutes and 1 km is not equal to 1000 m.)

Fill in the Blanks

Determine the denial to negate the truth value of the statement.

If m is a factor of 4, then m is a factor of 16.

Multiple Choice

Given the implication : 'if m > 10, then m > 8'. determine the false statement.

• Converse: if m > 8, then m > 10

• Inverse: if m < 10, then m < 8

• Contrapositive: if m < 8, then m < 10

• Contrapositive of the given implication is false.