Implications are "if-then" statements, which utilize propositional variables (like p and q). An implication is generally written as: p → q, which can be read aloud as "if p is true, then q is true", or "p implies q", or "if p, then q".
In an implication p → q, p is called the hypothesis and q is called the conclusion.
The hypothesis and conclusion can each be more sophisticated
propositional statements than just p and q, such as:
(p ∧ q) → r (just as an example.)
a. For the following statements given, assign a variable to each. The variable name can be any single-letter variable, so long as each statement has its own unique variable.
Using the variables above, write out the following statements using these variables.
Truth values for implications
For a statement of the form, if HYPOTHESIS, then CONCLUSION to be false,
it must be the case that the hypothesis is true, while the conclusion is false.
Otherwise, the statement is true.
The truth-table is as follows:
|p||q||p → q|
This might seem a little unintuitive.
The only way the result is false is when the hypothesis is true and the conclusion is false.
Think of this as: If the hypothesis is false, then our question is pointless anyway; if the hypothesis is false, then it doesn't affect the outcome of this implication because we can only derive the conclusion for a true hypothesis.
Complete the truth tables for the given compound expressions.
Given the predicates:
Rewrite the following statements using these predicates, and the symbols ∨ ∧ ¬ → (as appropriate).
The negation of the implication p → q is the statement p ∧ (¬q).
Notice that the negation is not also an implication.
We can see that they are logically equivalent via a truth-table:
|p||q||p → q||¬(p → q)||p ∧ (¬q)|
This is one of the tricky things about implications, so make sure to pay attention!
"If Bob has an 8:00 class today, then it is a Tuesday."
p: Bob has an 8:00 class today
q: it is a Tuesday
The negation is: ¬(p → q) ≡ p ∧ (¬q), which would be read as "Bob has an 8:00 class today, AND it is not Tuesday."
Write the negation of each of the following statements.
Do this in English terms (remember that the negation of "if p then q" is "p and not q")
For some implication p → q...
Another way of stating this is, suppose we have quantified statements,
P(n) is "n ends in a digit 2" and Q(n) is "n is divisible by 2", then...:
With the statements given, write either the converse, inverse, or contrapositive.
You can use quantifiers like Q(x) and P(x), or just p and q (or whatever variables seem appropriate, just define them.)