00:01
All right, so we want to write the negation of the following statements.
00:06
So for four, we're negating this statement for any integer in.
00:11
If then, so if it's negation, we have a p implies q.
00:18
When we negate it, that's going to turn to a p and not q.
00:23
So in that case, we're going to have for any integer n, it says p is going to stay the same so um n is a composite then and then not q so have an or so if we negate that that's going to become an and so we say and then both of those are negated n is odd and so have n is greater than two so n would be less than or equal to two that's for part a and then for part b of that of four it would be for all real numbers x and y um with x less than y there exists an in such that x is less than equal to n which is less than equal to y so we would say um there exists uh x and y which are real numbers still.
01:55
With x being greater than or equal to y, then it says there exists an integer to n, so it's a for all integers in, such that, so this would become now x is greater than n or n is greater than y.
02:30
For five, let the converse and contrapositive of a statement.
02:38
So the converse would be, that would be swapping the p and the q.
02:51
So therefore we would have for all real numbers.
03:01
If, so now we have q, x plus 1 is greater than 0, then now we have the p, then negative 1 is less than x, which is less than equal to 0.
03:17
For part b you want to do the contrapositive so we're going to swap them and then also negate them so we're going to have for all real numbers we have if now we're negating q so if x plus 1 is less to an equal to 0 then now negating p we'll have negative 1 is greater than equal to x or x is greater than zero...