Question 6 (a)

Expression to simplify:

¬(P v R) ∧ [ (P v Q) ∧ ¬Q ]

Method 1 — Algebraic simplification (laws of propositional algebra)

We use distributive, De Morgan, and simplification laws. Steps shown with plain text formulas.

Start:   ¬(P v R) ∧ [ (P v Q) ∧ ¬Q ]

Step 1: simplify the bracket (use distributive / identity):
        (P v Q) ∧ ¬Q
        = (P ∧ ¬Q) v (Q ∧ ¬Q)     (distribute ∧ over v)
        = (P ∧ ¬Q) v FALSE         (Q ∧ ¬Q is FALSE)
        = P ∧ ¬Q

So now expression becomes:
        ¬(P v R) ∧ (P ∧ ¬Q)

Step 2: apply De Morgan on ¬(P v R):
        ¬(P v R) = ¬P ∧ ¬R

Now:
        (¬P ∧ ¬R) ∧ (P ∧ ¬Q)
        = (¬P ∧ P) ∧ ¬R ∧ ¬Q     (reorder / assoc.)
        = FALSE ∧ ¬R ∧ ¬Q
        = FALSE

Therefore the whole expression simplifies to the contradiction FALSE (always false).
  

Final simplified result (Method 1): FALSE (a contradiction — expression is always false).

Method 2 — Truth-table verification (shows expression is always false)

We compute the value of the whole formula for all 8 combinations of P, Q, R.

Every row gives final value F → expression is always false. This agrees with Method 1.

Question 6 (b)

Given true statement: "If a creature is a whale, then it is a mammal."

Let W mean "creature is a whale" and M mean "creature is a mammal". The original statement is:

Original: W -> M   (This is given true)

Write the converse, inverse, and contrapositive, then evaluate their truth (with reasons/examples):

Converse:    M -> W
Inverse:     ¬W -> ¬M
Contrapositive: ¬M -> ¬W
  

Truth evaluation (with reasoning):

1. Original (W -> M): given true.
2. Converse (M -> W): usually false. Reason: many mammals are not whales (e.g., cats, dogs, humans, dolphins). Example: "a cat is a mammal" (M true) but "cat is a whale" (W false). So converse is false.
3. Inverse (¬W -> ¬M): usually false. Reason: "not a whale" does not imply "not a mammal" (a cat is not a whale but is a mammal). So inverse is false.
4. Contrapositive (¬M -> ¬W): logically equivalent to the original implication. If original is true, contrapositive is also true. Reason: if something is not a mammal, it cannot be a whale (all whales are mammals).

Which are logically equivalent? Original (W -> M) and Contrapositive (¬M -> ¬W) are logically equivalent. Converse and Inverse are equivalent to each other but not necessarily true.

Question 6 (c)

Argument:

1. If I like Mathematics, then I will study hard.        (P -> Q)
2. Either I study hard or I will fail.                  (Q v R)
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Therefore: If I fail then I do not like mathematics.     (R -> ¬P)
  

Symbols used:

P : I like Mathematics
Q : I will study hard
R : I will fail
  

We test validity by truth table: an argument is valid if whenever all premises are true the conclusion is also true (i.e. there is no row where premises true and conclusion false).

Look for any row where All premises = TRUE but Conclusion = FALSE. Row highlighted below is a counterexample.

Counterexample (row shown):

P = T, Q = T, R = T
P -> Q    = T
Q v R    = T
All premises = T ∧ T = T

Conclusion R -> ¬P:
R = T and ¬P = F  => R -> ¬P = F

Thus premises are true but conclusion is false in this row.
  

Conclusion: The argument is invalid. We found a situation (P true, Q true, R true) where premises are true but conclusion is false.

Short explanation in words: If you like math (P) and so you study hard (Q), and you still fail (R), the premises do not force the conclusion "if I fail then I do not like mathematics" — that conclusion can be false while premises true. Hence invalid.