Logical Reasoning – SS1 Mathematics Lesson Note (First Term)

Logical reasoning is the study of statements that are either true or false but not both.

• Truth value: T for a true statement and F for a false statement.
• Negation: the statement ~p, which means “not p”.
• Conditional statement: a statement of the form q ⇒ r, where q is the antecedent and r is the consequent.

Behavioural objectives

By the end of the lesson, students should be able to:

1. identify logical statements and tell them apart from questions, exclamations and commands;
2. write the negation of a given statement and show it in a truth table;
3. write a conditional statement in symbols (q ⇒ r) and name its antecedent and consequent;
4. find the truth value of a conditional statement;
5. write the converse, inverse and contrapositive of a conditional statement.

What is a logical statement?

A logical statement is a spoken or written declaration that is either true or false but not both.

• True statement: has truth value T, for example “Nigeria is an African country”.
• False statement: has truth value F, for example “The earth is conical in shape”.
• Symbols: logical statements are denoted by the letters p, q, r and so on.

What sentences are not logical statements?

Sentences that cannot be called true or false are not logical statements.

• Questions: “Who is he?”
• Exclamations: “What a lovely man!”
• Commands: “Take the pencil away.”
• Expressions of feeling: sentences that only show emotion.
• Incomplete sentences: “If I think of my family” gives no conclusion.

Which of these are logical statements?

What is negation?

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

• If p is true (T): ~p is false (F).
• If p is false (F): ~p is true (T).
• Truth table: a table that shows the truth values of p and ~p side by side.

What are examples of negation?

1. Let p be “Nigeria is a rich country”. Then ~p is “It is false that Nigeria is a rich country” or “Nigeria is not a rich country”.
2. Let r be 3 + 4 = 8. Then ~r is 3 + 4 ≠ 8.
3. Let q be “Isosceles triangles are equiangular”. Then ~q is “It is false that isosceles triangles are equiangular” or “Isosceles triangles are not equiangular”.

What are conditional statements?

A conditional statement is a compound statement of the form “if q then r”, written q ⇒ r.

• Simple statement: a single statement such as q or r.
• Compound statement: two simple statements combined, such as “if q then r”.
• Symbol ⇒: an operation that joins q and r.

How is q ⇒ r read?

• q implies r.
• If q then r.
• r if q.

What are the antecedent and the consequent?

In q ⇒ r, the antecedent is q and the consequent is r.

• Antecedent: the “if” part, the statement before the arrow.
• Consequent: the “then” part, the statement after the arrow.

Example: let q be “Femi is a brilliant student” and r be “Femi passed the test”.

• Compound statement: “If Femi is a brilliant student then Femi passed the test”.
• Antecedent: Femi is a brilliant student.
• Consequent: Femi passed the test.

When is a conditional statement true or false?

A conditional statement q ⇒ r is false only when q is true and r is false.

How do you find the truth value of a conditional statement?

1. Find the truth value of the antecedent.
2. Find the truth value of the consequent.
3. Read the answer from the truth table above.

Example: truth value of conditional statements

Let q be “I am a male” and r be “The sun will rise”. Take the speaker to be a male, so q is T and r is T.

• Statements (a), (c) and (d): all true.
• Statement (b): not true, because the antecedent is true and the consequent is false.

What are the converse, inverse and contrapositive?

The converse, inverse and contrapositive are three statements formed from a conditional statement q ⇒ r by swapping or negating its parts.

How do you write them?

1. Identify the antecedent q and the consequent r.
2. Apply the rule for the form you want.
3. Write the new statement in words, using “not” where there is a ~.

Example

Original: “If Femi is a brilliant student then Femi passed the test”.

• Converse: If Femi passed the test then Femi is a brilliant student.
• Inverse: If Femi is not a brilliant student then Femi did not pass the test.
• Contrapositive: If Femi did not pass the test then Femi is not a brilliant student.

A conditional statement and its contrapositive always have the same truth value.

Evaluation

1. Which of these is a logical statement: “Who is he?”, “Nigeria is an African country”, “Take the pencil away”?
2. If p is true, what is the truth value of ~p?
3. Let p be the statement “Ade is tall”. Write ~p in two ways.
4. In the statement “If Femi is a brilliant student then Femi passed the test”, name the antecedent and the consequent.
5. Explain why “If I am a male then the sun will not rise” is not true.

Assignment

Let p be “Ayo has determination” and q be “Ayo will succeed”. Answer questions 1 to 5, then questions 6 and 7.

For questions 1 to 5, choose the symbols that represent each statement.

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 ⇒ ~p
   • C. p ⇒ ~q
   • 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. Treat “he is humble and prayerful” as one statement.
   • (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