Logical Reasoning SS1 Mathematics Lesson Note

LOGICAL STATEMENTS

A logical statement is a declaration verbal or written that is either true or false but not both. A true statement has a truth value T. A false statement has a truth value F. Logical statements are denoted by letters p, q, r ……

Questions, exclamations, commands and expressions of feelings are not logical statements.

Ex: Which of the following are logical statements?

Nigeria is an African country        (Statement)

Who is he?                    (Not statement)

If I run I shall not be late             (Statement)

Japanese are hardworking people        (Statement)

What a lovely man!            (Not statement)

The earth is conical in shape        (Statement)

If I think of my family            (Not statement)

Take the pencil away            (Not statement)

 NEGATION

Given a statement p, the negation of p written ~p is the statement ‘it is false that p” or     “not p”

    If P is true,(T)  ~p is false(F)and if P is false(F)~p is true(T). 

The relationship between P and ~p is shown in a table called a truth table

 Eg I: Let P be the statement ‘Nigeria is a rich country’ then ~p is the statement ‘It is false that Nigeria is a rich country or ‘Nigeria is not a rich country’

 Eg II: Let r be the statement 3 + 4 = 8 then ~p is the statement 3 + 4 ≠ 8

Ex III: Let q be the statement ‘isosceles triangles are equiangular’ then ~q is the statement ‘it is false that isosceles triangles are equiangular or ‘isosceles triangles are not equiangular’.

CONDITIONAL STATEMENTS

Let q stand for the statement ‘Femi is a brilliant student’ and r stand for the statement ‘Femi passed the test’. One way of combining the two statements is ‘If Femi is a brilliant     student then Femi passed the test’ or ‘If q then r’

 The statement ‘If q then r’ is a combination of two simple statements q and r. It is called a compound statement.

Symbolically, the compound statement can be written as follows: ‘If q then r’ as q ⇒ r

    The statement q ⇒ r is real as 

    q implies r or 

    if q then r or 

    q if r

 The symbol ⇒ is an operation. In the compound statement q ⇒ r, the statement q is called the antecedent while the sub statement r is called the consequence of q ⇒ r.

Ex: If q is the statement ‘I am a male’ and r is the statement ‘The sun will rise’

        Consider the statements.

If I am a male then the sun will rise

If I am a male then the sun will not rise

If I am not a male then the sun will rise

If I am not a male then the sun will not rise

The statement (a), (c) and (d) are all true but b is not true because the antecedent is true and the consequent is false.

ASSIGNMENT

P is the statement ‘Ayo has determination and q is the statement ‘Ayo will succeed’. Use this information to answer these questions.

Which of these symbols represents these statements?

1. Ayo has no determination. A. P ⇒ q      B.   ~ p ⇒ q          C.      ~ p

2. If Ayo has no determination then he won’t succeed. A. ~p ⇒~ q     B. p ⇒~ q        C.  p ⇒ q        D.   p ⇒~ q

3. If Ayo won’t succeed then he has no determination. A. ~q ⇒ p      B.    ~q ⇒~q        C.   ~q ⇒ p      D. q ⇒ p

4. If Ayo has determination then he will succeed. A. ~p ⇒ q     B. ~p ⇒~ q       C.  ~q ⇒~ p     D. p ⇒ q

5. If Ayo has no determination then he will succeed. A. ~p ⇒ q     B.  ~q ⇒~ p       C. ~p      D. ~p ⇒~ q

  6. Write down the inverse, converse and contrapositive of each of these statements.

(i)    If the bank workers work hard they will be adequately compensated.

(ii)    If he is humble and prayerful, he will meet with God’s favour.

(iii)    If he sets a good example, he will get a good followership.

7. Find the truth value of these statements. a) If 11 > 8 then -1< -8 b) If 3 + 4 ≠ 10 then 2 + 3 ≠ 5