Consider the open sentence \\(P(x,y,z) : (x-1)^2 + (y-2)^2 + (z-2)^2 > 0\\) wher
ID: 2962122 • Letter: C
Question
Consider the open sentence
(P(x,y,z) : (x-1)^2 + (y-2)^2 + (z-2)^2 > 0)
where the domain of each of the variables x, y and z is R (real numbers)
(a) Express the quantified statement %u2200x %u2208 R, %u2200y %u2208 R, %u2200z %u2208 R, P(x,y,z) in words.
(b) Is the quantified statement in (a) true or false? Explain.
(c) Express the negation of the quantified statement in (a) in symbols.
(d) Express the negation of the quantified statement in (a) in words.
(e) Is the negation of the quantified statement in (a) true or false? Explain.
Explanation / Answer
(a) Express the quantified statement %u2200x %u2208 R, %u2200y %u2208 R, %u2200z %u2208 R, P(x,y,z) in words...UP LOAD NOT CLEAR ...WHAT DOES THIS MEAN ?
ASSUMING P IS THE STATEMENT ...
FOR X,Y,AND Z REAL NUMBERS , THE SUM OF SQUARES OF X-1 , Y-2 AND Z-2 IS POSITIVE ..........
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(b) Is the quantified statement in (a) true or false? Explain.
FALSE .......AT X=1, Y=Z=2 , WE GET P=0 NOT GREATER THAN ZERO OR POSITIVE ...
EXCEPT THIS POINT FOR ALL OTHER REAL VALUES OF X,Y,Z THIS IS TRUE ...
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(c) Express the negation of the quantified statement in (a) in symbols.
[X-1]^2 + [Y-2]^2 + [Z-2]^2 < = 0 ...FOR ALL REAL X,Y,Z....
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(d) Express the negation of the quantified statement in (a) in words.
FOR X,Y,AND Z REAL NUMBERS , THE SUM OF SQUARES OF X-1 , Y-2 AND Z-2 IS NOT POSITIVE ...OR...IT IS NEGATIVE OR ZERO ........
(e) Is the negation of the quantified statement in (a) true or false? Explain.
FALSE ......
SUM OF SQUARES IS ALWAYS NON NEGATIVE .. THAT IS IT COULD BE ZERO AS .AT X=1, Y=Z=2 , OR POSITIVE AT ANY OTHER VALUE OF X OR Y OR Z ..
SO THIS IS NEVER TRUE ..
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