a The propositions P and 3aPx are logically equivalent b The

(a) The propositions P and 3a-P(x) are logically equivalent. (b) There exist integers a, y such that 89x+101y 1.

Solution

a)

True

First statement in words is :

For all x P(x) is true

In other words this means

There does not exist x so that negation of P(x) is true

which is the second statement.

b)

True,

Because 89 and 101 are prime numbers so their gcd is 1 and hence by Euclid algorithm we can find x,y so that

89x+101y=gcd(89,101)=1