Statement is a sentence that has a logical value (true or false)
From statements pand qwe can derive the following statements:
-‘pand q’ (symbolic notation: ‘p∧q’)
-‘por q’ (symbolic notation: ‘p∨q’)
-‘If pthen q’, equivalently: ‘pimplies q’ (symbolic notation: ‘p→q’ or ‘p⇒q’).
-‘Not p’ (symbolic notation: ¬p)
For an implication ‘p→q’ we have two related implications:
-the converse, ‘q→p’
-the contrapositive, ‘¬p→ ¬q’
An implication and its contrapositive always have the same logical value.
Equivalence. ‘p↔q’ means the same as ‘(p→q)∧(q→p)’. The logical value of ‘p↔q’ is
true if pand qhave the same logical value (both true or both false). The statements ‘p
if and only if q’, ‘piff q’ and ‘pis equivalent to q’ all mean ‘p↔q’.
Rules for negations. The following pairs of statements always have the same logical value:
-‘¬(p∨q)’ and ‘¬p∧ ¬q’
-‘¬(p∧q)’ and ‘¬p∨ ¬q’
-‘¬(p→q)’ and ‘p∧ ¬q’
-‘¬¬p’ and ‘p’
The first two rules above are called the deMorgan Laws.
Statements with quantifiers (for every.../there exists...). Below, p(x) is a statement depend-
ing on a variable x.
-‘For each x,p(x)’ (symbolic notation: ‘∀xp(x)’)
-‘There exists xsuch that p(x)’ (symbolic notation: ‘∃xp(x)’)
Negations of statements with quantifiers. The following pairs of statements have the same
logical value:
-‘¬∀xp(x)’ and ‘∃x¬p(x)’
-‘¬∃xp(x)’ and ‘∀x¬p(x)’
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