3 12 points Express each of these statements using quantifie

3. (12 points) Express each of these statements using quantifiers and the given predicates, along with arithmetic operations (e.g. 2,2x). Then form the negation of the statement, so that no negation operator is to the left of a quantifier. Next, express this negation you formed in simple and precise English. (a) Statement: For all real numbers z, ifra21, then z > b) Statement: For all integers a, b, c, if a -b is even and b-cis even, then a -c is even. (c) There is a real number with no reciprocal. Allowed predicates: G(a, b) = \"a > b\", Eq(a, b) = \"a = b\". Allowed predicate: Eq(a, b) = \"a = b\". Reminder: the reciprocal of a real number a is a real number b such that ab = 1. Allowed predicates: Eq(a, b) = \"a = b\" (d) There is a real number z, such that for all real numbers y, 2z+ y = 5. Allowed predicates: Eq(a, b-\"a-b

Solution

Hi,
c.
given there is a real number with no reciprocal
we can write the same in quanitifiers as
yR,!xR Eq(xy,1) - this means, for all y in R, there doesnt exist a x such that xy is 1
Now, forming a negation, we have to flip the for all and there exists quantifiers
yR,xR Eq(xy,1)
the above statement means the exact opposite i.e
For all y in R, there exists x in R such thatxy=1

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