Question

Let the predicates P,T, and E be defined below. The domain is the set of all positive integers.

P(x): x is odd

T(x, y): 2x < y

E(x, y, z): xy - z

Indicate whether each logical expression is a proposition. If the expression is a proposition, then give its true value and show your work. If the expression is not a proposition, explain why no.

1(a) P(5)

1(b) ¬P(x)

1(c) T(5, 32)

1(d) ¬P(3) V ¬T(5, 32)

1(e) T(5,10) → E(6, 3, 36)

Answer 1

1(a) Yes it is a proposition because,In P(X) x is an odd number and we have x=5 which is an odd number so it also gives true value.

1(b) (not)P(X) means x is not an odd number.But "x" is a variable so it is not proposition.

1(c) In T(x,y) 2x <y, we have x=5 and y=32 , so we get 2x=2*5=10 and y=32

therefore 10<32 so it is true so it a proposition and it gives true value.

1(d) In (not)P(3) means 3 is not odd number but it is an odd number so it will give false result,now in (not)T(5,32) means 10 is not less than 32 which is also false and as there are the disjunction sign between both of them so false or false gives false result.So it is a proposition. If either condition is true then it gives true result for example if we have (not)p(4) V (not)t(5,32) then it gives true result.

1(e) Yes it is proposition and In T(5,10) → E(6, 3, 36) ,it will gives false only when  T(5,10) is true and  E(6, 3, 36) gives false otherwise it always gives true result let's check

In T(5,10) 2*5=10<10 which is false and as T(5,10) gives false result then T(5,10) → E(6, 3, 36) always gives true result.