Question

Logic Discrete Maths Question 3 & 4

3. [6 marks: 3 marks for steps, 3 marks for labels] Simplify the following statement using the laws and axioms of logic. Clearly state which law or axiom has been used at each step 4. [4 +4-8 marks] Given the following statements The student is in the esports club or in the aquatic club. If they are in the esports club then they do not get free access to the pool. The student does get free access to the pool. Therefore the student is in the aquatic club a) Convert these statements into propositional logic statements (1 mark per statement) b) Without using a truth table, show that this argument is valid. This could be done using contradiction or the laws/axioms of algebra (note the latter is a long process though)

Answer 1

(7a v b) ^ (a v b) ^ 7a
Applying distributive law
= ((7a ^ a) v (b ^ a) v (7a ^ b) v (b ^ b)) ^ 7a
Applying Complement law
= ((F) v (b ^ a) v (7a ^ b) v (b ^ b)) ^ 7a
Applying identity law
= ((b ^ a) v (7a ^ b) v (b ^ b)) ^ 7a
Applying idempotent law
= ((b ^ a) v (7a ^ b) v (b)) ^ 7a
Applying distributive law
= ((b ^ a ^ 7a) v (7a ^ b ^ 7a) v (b ^ 7a))
Applying Complement law
= ((b ^ F) v (7a ^ b ^ 7a) v (b ^ 7a))
Applying dominant law
= ((F) v (7a ^ b ^ 7a) v (b ^ 7a))
Applying identity law
= ((7a ^ b ^ 7a) v (b ^ 7a))
Applying idempotent law
= ((7a ^ b) v (b ^ 7a))
Applying idempotent law
= (7a ^ b)

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