TMUA Practice Question — Worked Solution

P2·Logic & Quantifiers·6.0Mock 0 · P2 · Q5

logical equivalencecontrapositiveonly ifnecessary conditionnegation of andoption-table

In this question,

a

and

b

are positive integers.Consider the following theorem:If

a

and

b

are both odd, then

a^{2} + b^{2}

is not a perfect square.

(*)

Which of the following statements, taken individually, is/are equivalent to

(*)

a^{2} + b^{2}

is a perfect square, then

a

and

b

are both even.

a

and

b

are both odd only if

a^{2} + b^{2}

is not a perfect square.

III

A necessary condition for

a^{2} + b^{2}

to be a perfect square is that at least one of

a

and

b

is even.

	Statement I is equivalent to $(*)$	Statement II is equivalent to $(*)$	Statement III is equivalent to $(*)$
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No

Discussion

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	Statement I is equivalent to $(*)$	Statement II is equivalent to $(*)$	Statement III is equivalent to $(*)$
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No

	Statement I is equivalent to $(*)$	Statement II is equivalent to $(*)$	Statement III is equivalent to $(*)$
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No

	Statement I is equivalent to $(*)$	Statement II is equivalent to $(*)$	Statement III is equivalent to $(*)$
A	Yes	Yes	Yes
B	Yes	Yes	No
C	Yes	No	Yes
D	Yes	No	No
E	No	Yes	Yes
F	No	Yes	No
G	No	No	Yes
H	No	No	No