Answer to Question #262201 in Discrete Mathematics for Moomina

Question

Write the following logical arguments as predicate expressions, defining the predicates used and domains of

variables. For each argument, mention the inference rules used in each step. [6 marks]

a) “Asim, a student in this class, knows how to write programs in Java. Everyone who knows how to write

programs in Java can get a high-paying job. Therefore, someone in this class can get a high-paying job.”

b) “Somebody in this class enjoys whale watching. Every person who enjoys whale watching cares about

ocean pollution. Therefore, there is a person in this class who cares about ocean pollution.”

c) “Each of the 50 students in this class has a laptop. Everyone who has a laptop can use a word processing

program. Therefore, Asim, a student in this class, can use a word processing program.”

d) “Everyone in Karachi lives within 50 miles of the ocean. Someone in Karachi has never seen the ocean.

Therefore, someone who lives within 50 miles of the ocean has never seen the ocean.”

Answer 1

Part a

Let p(x) be "x is a student", q(x) means "x knows how to code in java", r(x) means "x can get a high-paying job"

We have next statements: $\exists x(p(x)\land q(x))$ , $\forall x(q(x)\to r(x))$

we have to check whether the statement $\exists x:(p(x)\to r(x))$ is true

lets assume it is false, then $\forall x: (p(x)\to \lnot r(x))$ , but from the first two statements we can tell that b

$\exists x(p(x)\to r(x))$ . So, we came to contradiction, which means our statement is false, which means there is someone in the class who can get a high-paying job

Part b

Let P(x) = x cares about ocean pollution, C(x) = x is in the class, W(x)= x enjoys whale watching. Then the premises are $\forall x(W(x) \to P(x)), \exists x(W(x)\land C(x))$ and the conclusion is $\exists x(C(x)\land P(x))$

Part c

Let C(x) = x is in the class, P(x) = x owns a personal computer, W(x). x can use a word processing program, and Z = Asim.

Then the premises are $\forall x(C(x) \to P(x)), \forall x(P(x) \to W(x)),C(Z)$ , and the conclusion is W(Z). 

Part d

Let J (x) = x lives in Karach, 0 (x) = x lives within 50 miles of the ocean, S(x) = x has seen the Ocean. Then the premises are $\forall x(J (x \to 0(x)), \exists x(J(x) \land ¬ S(x))$ , and the conclusion is $\exists x(O(x) \land ¬ S(x))$