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Review for Mathematics: A Human Endeavor - Lecture Slides | MAT 1160G, Study notes of Mathematics

Material Type: Notes; Class: Mathematics: A Human Endeavor; Subject: Mathematics; University: Eastern Illinois University; Term: Spring 2005;

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

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Sec 3.3 Review
vAconditional statement uses implication () or if...else
vpqis false only when pis true and qis false.
vpqis equivalent to (pq)
vThe negation of pqis (p q)
vWe can use Truth Tables to show two conditional expressions are
equivalent (their truth values will be the same)
vAtautology is a statement which is always TRUE.
vCircuits in series correspond to conjunctions (ands)
vCircuits in parallel correspond to disjunctions (ors)
vSome circuits can be simplied.
© 2005–09, N. Van Cleave 1
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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

  1. p ∧ (q ∨ ∼ p)
  2. (p ∨ q) ∧ (∼ p ∧ ∼ q)
  3. [(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.

  1. The car is a Chevy. The car is a Toyota.
  2. Elvis is alive. Elvis is dead.
  3. The animal has four legs. The animal is a dog.
  4. The cake is chocolate. The cake has two layers.
  5. The clock is broken. The clock has the right time.
  6. The math class meets at noon. The math class lasts 50 minutes.
  1. The number is an integer. The number is irrational.
  2. The punch is pink. The punch has juice in it.
  3. President Bush is a Republican. President Bush is a Democrat.
  4. The sofa is soft. The sofa is blue.
  5. The plant is blooming. The plant is dead.
  6. The dog ate my homework. The dog bites.
  7. That rock is igneous. That rock is sedimentary.
  8. 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.