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To prove x z, it suffices to show that if x is true, then z must be true. If x is true, from x y, we know that y is true. Now, from y z and y is true, we conclude that z is true. Hence x z. The statement (x implies y) is defined to be the statement ( (not x) or y).
If $X\implies Y$ and $X\implies Z$, does that mean that $Y\implies Z$? I think it does, but can anyone show this as a proof? Thanks
Conditional statements are also called implications. An implication is the compound statement of the form “if , then .” It is denoted , which is read as “ implies .” It is false only when is true and is false, and is true in all other situations.
- The Basics
- A Mis-Understanding
- Trick 2: Assume ¬Q\Neg Q. Then Prove ¬P\Neg p
- Trick 3: Proof by Contradiction
- Which Trick Should I Apply?
Many of the proofs we will encounter in this course will be required to prove logical statements of the form: which is logically equivalent to the form P⟹Q (where in this case P stands for blah and Q stands for blah'). In this section we will see many ways to prove such a statement. To do that we will consider the following exercise throughout: To ...
Here is a common mistake when one encounters P⟹Q. Looking at Exercise 1, one might say "I know someone who is active and has brown eyes," e.g. this guy The above happens because of a mis-understanding of the logical expression P⟹Q. If P is false then logically the statement P⟹Q is true. Hence, we "only" need to worry about the case when P is true a...
The next trick is to use the fact that ¬Q⟹¬P is logically equivalent to P⟹Q: Next, we use this trick to solve Exercise 1.
Proof by contradiction is a very general (and useful) proof technique, where the idea is to assume the negation of what you're trying to prove and then come up with a contradiction (i.e. where you prove both a statement and its negation to be true). In the context of proving P⟹Q, this generally takes the following form: We now use this trick to sol...
You might have noted that the three solutions to Exercise 1 are similar. A natural question to ask is if there is a way to decide when one should use a particular trick. Unfortunately, the answer is that there is no mechanical way to do this. The more proofs you write the better you will get at picking the "correct" trick. (By correct I mean that s...
Apr 1, 2023 · Consider the implication: if n is an odd integer, then 5n+1 is even. Write the converse, inverse, contrapositive, and biconditional statements. More importantly, we will also discover how to determine the truth value for various implications using truth tables. For example,
The answer is (E). You gain this by resorting to the definition of implication, $x\implies y \equiv \neg x \vee y $ and the usual logic algebra.
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For example, the statement about a real number x ‘∃y ∈ R s.t. y2 = x’ implies the statement ‘x ≥ 0’. (try writing it down in words without any symbols and see if you agree). The statements A and B are called equivalent if they imply each other. In this case we say A holds if and only if B holds.
