Quantifiers An important mathematical concept Is this true
- Slides: 62
Quantifiers An important mathematical concept
• Is this true? Yes No Something else
• Is this true? Yes No Something else
• Is this true? Yes No Something else That’s why Epp always gives domain!
• Yes No Something else
• Yes No Something else
“For all” •
“For all” •
“For all” •
“For all” •
“For all” • True False Something else
“For all” • True False Something else
Nesting quantifiers •
Nesting quantifiers •
Nesting quantifiers •
Nesting quantifiers •
Nesting quantifiers •
Alternating nested quantifiers •
Alternating nested quantifiers •
Finding domains •
Finding domains •
Finding domains •
Finding domains •
Finding domains •
Finding domains •
Finding domains •
Some notation •
Examples • True False Something else
Examples • True False Something else
Examples • True False Something else Make sure you understand the order of parentheses ( (, ) ) and square brackets ([, ])!
Examples • Statement: Every even integer above 2 can be expressed as the sum of two primes. True False Something else
Examples • True False Something else
Existential statements •
Proving Existential Statements true •
Proving Existential Statements false •
Proving Existential Statements false •
Proving Universal Statements true or false • False: Similar to proving an existential statement true. • The witness this is this case is known as the counterexample. • True: Similar to proving an existential statement false. • Proving universal statements true is pretty much all of math.
Negated Quantifiers • It is not the case that Alice comes to all office hours. • There is an office hour that Alice does not come to. Equiv. Not equiv. Something else
Negated Quantifiers • It is not the case that Alice comes to all office hours. • There is an office hour that Alice does not come to. Equiv. Not equiv. Something else
Negated Quantifiers • We can therefore reach the following logical equivalences:
Negating nested quantifiers •
Negating nested quantifiers •
Negating nested quantifiers •
Negating nested quantifiers •
Negating nested quantifiers •
Another example •
Another example •
Another example •
Another example •
Another example •
Another example •
Negating the statement • How do we negate an implication?
Negating implications •
Back to our example •
Back to our example •
Back to our example •
Back to our example •