let Q(x) denote \"x is even,\" P(x) denote \"x is prime,\" R(x) denote \"x is di

Question

let Q(x) denote "x is even," P(x) denote "x is prime," R(x) denote "x is divisble by 6," G(x) denote x<= 5 and L(x,y) denote x<y. Determine what each of the following propositions asserts and whether it is true or false. The domain under discussion is the natural numbers.

a. x:(R(x)^P(x))

b. (x:R(x))^(x:P(x))

c.x:(G(x) -> P(x))

d.x:(P(x) -> ~(Q(x)) )

e. (x: P(x)) -> x: ~(Q(x))

f. (x: P(x)) -> x: (P(x) ^ R(x) )

g.x: (P(x) -> (P(x) ^ R(x) ) )

h. x:(R(x) -> y:(L(x,y) ^ (R(y) ) )

i.x:y: (L(x,y) ^ L(y,x) )

Explanation / Answer

a. x:(R(x)^P(x)) ---- true

b. (x:R(x))^(x:P(x)) -----true

c.x:(G(x) -> P(x)) ---- false

d.x:(P(x) -> ~(Q(x)) ) ----- true   

e. (x: P(x)) -> x: ~(Q(x)) ------ false

f. (x: P(x)) -> x: (P(x) ^ R(x) ) -----false

g.x: (P(x) -> (P(x) ^ R(x) ) ) ---false

h. x:(R(x) -> y:(L(x,y) ^ (R(y) ) ) ---- true

i.x:y: (L(x,y) ^ L(y,x) ) -------- false

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