Download Review for Mathematics: A Human Endeavor - Lecture Slides | MAT 1160G and more Study notes Mathematics in PDF only on Docsity!
Sec 3.3 Review
v A conditional statement uses implication (→) or if...else
v p → q is false only when p is true and q is false.
v p → q is equivalent to (∼ p ∨ q)
v The negation of p → q is (p ∧ ∼ q)
v We can use Truth Tables to show two conditional expressions are equivalent (their truth values will be the same)
v A tautology is a statement which is always TRUE.
v Circuits in series correspond to conjunctions ( and s)
v Circuits in parallel correspond to disjunctions ( or s)
v Some circuits can be simplied.
Rewrite as Boolean Expressions and Simplify
~p p
~p
p
r
p q
p
Simplify and draw circuits
- p ∧ (q ∨ ∼ p)
- (p ∨ q) ∧ (∼ p ∧ ∼ q)
- [(p ∨ q) ∧ r] ∧ ∼ p
Sec 3.4 More on the Conditional:
Converse, Inverse, and Contrapositive
Direct Statement p → q If p, then q Converse q → p If q, then p Inverse ∼ p →∼ q If not p, then not q Contrapositive ∼ q →∼ p If not q, then not p
Let p = ìthey stayî and q = ìwe leaveî Direct Statement (p → q):
Converse :
Inverse :
Contrapositive :
Equivalent Conditionals
Direct Converse Inverse Contrapositive p → q q → p ∼ p → ∼ q ∼ q → ∼ p p q ∼ p ∨ q
T T T T
T F F T
F T T F
F F T T
→ 4 is equivalent to ∼ ∨ 4
∼ ∨ 4 ≡ → 4
∨ 4 ≡ ∼ → 4
Tricky Questions
For p ∨ q, write each of the following:
Direct Statement :
Converse :
Inverse :
Contrapositive :
Rewrite as if...then statements & give some alternatives for
You'll be sorry if I go.
Today is Thursday only if yesterday was Wednesday.
All nurses wear white shoes.
A stitch in time saves nine.
Rolling stones gather no moss.
Birds of a feather ock together.
Consistent or Contrary?
Two statements about the same object are: consistent ó if they are both true. contrary ó if they cannot both be true.
- The car is a Chevy. The car is a Toyota.
- Elvis is alive. Elvis is dead.
- The animal has four legs. The animal is a dog.
- The cake is chocolate. The cake has two layers.
- The clock is broken. The clock has the right time.
- The math class meets at noon. The math class lasts 50 minutes.
- The number is an integer. The number is irrational.
- The punch is pink. The punch has juice in it.
- President Bush is a Republican. President Bush is a Democrat.
- The sofa is soft. The sofa is blue.
- The plant is blooming. The plant is dead.
- The dog ate my homework. The dog bites.
- That rock is igneous. That rock is sedimentary.
- That bird is a robin. That bird is blue.
True or False?
A biconditional is true only when both statements are true or
both statements are false.
True or False : 5 = 9 4 if and only if 8 + 2 = 10
True or False : Clinton was president IFF Carter wasn't president.
True or False : IBM sells computers IFF Pizza Hut sells Big Macs.
True or False : 8 + 7 15 IFF 3 5 9.
In Summary
∼ p negation of p truth value is opposite of p
p ∧ q conjunction true only when both p and q are true
p ∨ q disjunction false only when both p and q are false
p → q conditional false only when p is true and q is false
p ↔ q biconditional true only when p and q have the same truth value.
