Logical Reasoning: Worked Examples (Medium)

Worked example 1

1. Label the parts: p is 'x = 3' and q is 'x² = 9'; the implication is 'if p then q'.
2. The converse swaps the parts to 'if q then p': 'If x² = 9, then x = 3.'
3. Test the converse: x² = 9 gives x = 3 or x = −3.
4. Take x = −3: then x² = 9 is true but x = 3 is false, so the converse fails.

Worked example 2

1. Find the truth value of p: 15 ends in 5, so it is odd; p is true.
2. Form ~p, the negation of p: since p is true, ~p is false.
3. Find the truth value of q: 15 = 3 × 5, so 15 is divisible by 3; q is true.
4. Combine: '~p or q' = 'false or true'; an 'or' statement is true when at least one part is true, so the result is true.

Worked example 3

1. Recognise the argument form: 'All A are B; C is A; therefore C is B.'
2. Match: A = squares, B = shapes with four equal sides, C = PQRS.
3. Apply the form: since PQRS (C) is a square (A), it must have property B.
4. Write the conclusion: 'PQRS has four equal sides.' The form is a valid deductive pattern.

Worked example 4

1. An implication 'if p then q' has contrapositive 'if not q then not p'.
2. Here p: the shape is a square; q: it has four right angles.
3. Contrapositive: 'If a shape does not have four right angles, then it is not a square.'
4. The original statement is true, and a contrapositive always shares the same truth value, so it is true.

Worked example 5

1. p: '6 is a prime number' is false, since 6 = 2×3.
2. q: '6 is an even number' is true, so ~q is false.
3. 'p and (~q)' is true only when both parts are true.
4. Here p is false and ~q is false, so 'p and (~q)' is false.

Worked example 6

1. Premise 1 states a rule true for every multiple of 10.
2. Premise 2 says 250 is a multiple of 10.
3. Applying the general rule to this particular case, 250 must end in the digit 0.
4. The conclusion follows directly from the two premises, so the argument is valid.

Worked example 7

1. Original: p → q, where p: 'a number is divisible by 6', q: 'a number is divisible by 3'.
2. Contrapositive: ~q → ~p, i.e. 'If a number is not divisible by 3, then it is not divisible by 6.'
3. A contrapositive always has the same truth value as the original implication.
4. The original statement is true (every multiple of 6 is a multiple of 3), so the contrapositive is also true.

Worked example 8

1. Premise 1 gives a general rule about all Form 5 students.
2. Premise 2 states that Aiman belongs to this group.
3. Applying the rule to Aiman gives the conclusion: Aiman sits for the SPM examination.
4. This follows the valid pattern 'All A are B; x is A; therefore x is B', so the argument is valid.

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)