Suppose that P(x) and Q(x) are mathematical statements about some object x, and X is some set. (a) Write down a roadmap (as in lectures, I mean by this an outline of the structure a proof could have including the starting point and conclusion but omitting the details) for proving the following statement is true. For all x E X, P(x) ⇒ Q(x). Let S be the statement: For all x, y € R, if x + y = 0 then xy ≤ 0. (b) Decide whether the statement S is true or false, giving a proof or counterexample as appropriate. (c) Write down the statement obtained by replacing the implication in S by its converse. Decide whether this new statement is true or false, giving a proof or counterexample as appropriate. (d) Write down the statement obtained by replacing the implication in S by its contrapositive. Decide whether this new statement is true or false, giving a proof or counterexample as appropriate.
Suppose that P(x) and Q(x) are mathematical statements about some object x, and X is some set. (a) Write down a roadmap (as in lectures, I mean by this an outline of the structure a proof could have including the starting point and conclusion but omitting the details) for proving the following statement is true. For all x E X, P(x) ⇒ Q(x). Let S be the statement: For all x, y € R, if x + y = 0 then xy ≤ 0. (b) Decide whether the statement S is true or false, giving a proof or counterexample as appropriate. (c) Write down the statement obtained by replacing the implication in S by its converse. Decide whether this new statement is true or false, giving a proof or counterexample as appropriate. (d) Write down the statement obtained by replacing the implication in S by its contrapositive. Decide whether this new statement is true or false, giving a proof or counterexample as appropriate.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Suppose that P(x) and Q(x) are mathematical statements about some object x, and X
is some set.
(a) Write down a roadmap (as in lectures, I mean by this an outline of the structure a
proof could have including the starting point and conclusion but omitting the
details) for proving the following statement is true.
For all x E X, P(x) ⇒ Q(x).
Let S be the statement:
For all x, y € R, if x + y = 0 then xy ≤ 0.
(b) Decide whether the statement S is true or false, giving a proof or counterexample
as appropriate.
(c) Write down the statement obtained by replacing the implication in S by its
converse. Decide whether this new statement is true or false, giving a proof or
counterexample as appropriate.
(d) Write down the statement obtained by replacing the implication in S by its
contrapositive. Decide whether this new statement is true or false, giving a proof
or counterexample as appropriate.
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