LOGIC proving/disproving general laws

[dburton]

One of the following general laws is true, the other false. Prove the true one. Find a counter example to the other.(We must do this in writing out each statement, no truth tables or anything.

(a) If X implies Y, then the disjunction of X or Y is equivalent to X.
(b) If X implies Y, then the disjunction of X or Y is equivalent to Y.

I don't know, could you help me write out each step to prove/disprove?

 

[KingOrdo]

One of the following general laws is true, the other false. Prove the true one. Find a counter example to the other.(We must do this in writing out each statement, no truth tables or anything.

(a) If X implies Y, then the disjunction of X or Y is equivalent to X.
(b) If X implies Y, then the disjunction of X or Y is equivalent to Y.

I don't know, could you help me write out each step to prove/disprove?

Okey. We have as an assumption "X implies Y". That means that either (1) X is false and Y is false, (2) X is false and Y is true, or (3) X is true and Y is true (because "X implies Y" is false if and only if X is true and Y is false). "[T]he disjunction of X or Y" (this is phrased a little funny) is false for (1) and true for (2) and (3). So we need to figure out if X or Y is similarly false for (1) and true for (2) and (3). That will be your answer. We see that Y satisfies these criteria, so (b) is the correct answer. A counterexample to (a) is the case where X is false and Y is true. Then "X implies Y" is true, but "the disjunction of X or Y" (true) is not equivalent to X (false).