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Dr. Math FAQ:
Browse High School Logic
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Selected answers to common questions:
- Truth of a Biconditional Statement [11/08/2005]
Let p represent x = 0, and let q represent x + x = x. Write the
biconditional p <-> q in words. Decide whether the biconditional is true.
- Truth of the Contrapositive [06/07/2003]
The inverse of a statement's converse is the statement's
- Truth Tables: a AND b [09/10/2001]
What are truth tables?
- Truth Tables: And, Or, Implies, Not [06/10/2001]
Can you give me an explanation of truth tables, with examples?
- Truth Tables of Boolean Variables [03/09/2001]
Given two Boolean variables, A and B, what are the sixteen possible truth
functions in table form?
- Two Mathematicians: Factoring Logic [03/24/2003]
Two mathematicians are each assigned a positive integer. They are told
that the product of the two numbers is either 8 or 16. Neither knows
the other's number...
- Two Mathematicians Problem [05/18/1998]
One mathematician is give the sum of integers X and Y, and another is
given their product... what are the numbers?
- Two Turkeys [10/07/1997]
... How much did each turkey weigh?
- Two- vs. Many-Valued Logic [06/19/2001]
Are the perhaps practical instruments of "fuzzy logic" and not at least
"three- or many-valued logic" for pragmatic use only?
- Uncountable Infinitude, Illogically Concluded [11/21/2010]
If a rational number can be found between any two irrationals, and the set of
irrationals are uncountably infinite, does that mean that the rationals are also
uncountable? Doctor Peterson points up the flaw in a student's assumption about what
to conclude from a failed mapping.
- Understanding the Transitive, Reflexive, and Symmetric Properties [06/30/2008]
Decide if the relation 'is not equal to' is a)transitive, b)reflexive,
and c)symmetric with regard to the counting numbers.
- Union and Intersection of Empty Family of Sets [03/20/2003]
In what sensible way can we define the union and intersection of an
empty family of sets?
- Unions and Intersections: Proving Sets [10/17/1999]
How can I verify a proof of the statement A - (B union C) = (A - B)
intersect (A - C)?
- Unknown Numbers and a Venn Diagram [11/26/2001]
The GCF of two numbers is 20 and the LCM is 840. One of the numbers is
120. Explain how to find the other number and use the Venn diagram method
- Upside-down A's and Sideways E's [09/19/2001]
What is an upside-down A? a sideways E? Also, what do R and J stand for?
- Vacuous Cases, Empty Sets, and Empty Functions [04/10/2004]
I am having difficulty understanding 'vacuous' situations such as if A
is an empty set and B is a non-empty set, then why is there one
function mapping A to B (the empty function) but no function mapping B
- Valid Arguments [12/17/2002]
What are the real-life applications for valid arguments?
- Venn Diagram - Choose One of Three Options [01/24/1999]
Members of a computer class choose at least one of three options. How
many are taking just one? ... Use a Venn diagram.
- Venn Diagram: Goops, Gorps, Gorgs [09/19/2002]
Every Goop is a Gorp. Half of all Gorgs are Gorps. Half of all Gorps
are Goops. There are 40 Gorgs and 30 Goops. No Gorg is a Goop. How
many Gorps are neither Goops nor Gorgs?
- Venn Diagram of Natural Numbers [09/22/1999]
How can I construct a Venn diagram comparing the numbers 1 through 100 in
these 4 areas: odd, even, composite and prime?
- Venn Diagram of Our Number System [12/13/2002]
I don't know how to include complex numbers that consist of a real
part and an imaginary part. Can you please diagram this for me?
- Venn Diagram to Classify Quadrilaterals [01/02/2003]
I am looking for a Venn diagram that will accurately display the
relation among trapezoids, parallelograms, kites, rhombi, rectangles,
- Venn Diagram: Two Possibilities [01/14/2003]
The science club advisor asked club members what science courses they
liked. Eighteen members said they liked physics, 17 liked chemistry,
and 10 liked biology. However, of these, 9 liked physics and
chemistry, 4 liked biology and chemistry, 2 liked physics and biology,
and 2 liked all three. How many science club members were interviewed?
- What Color Hat am I Wearing? [4/17/1996]
Three students close their eyes, and the teacher puts a hat on each of
their heads (hiding the other two hats)...
- What is a Property? [11/29/2001]
I understand Undefined and Defined terms and Axioms and Theorems, but
what exactly is a Property? Is it the same thing as a Theorem? Also what
is a Law?
- What is a Venn Diagram? [02/26/1998]
What is a Venn Diagram? What is its use, definition, and what does it
- Where is the Arsenic? [03/12/2002]
You place six jars (right to left: coffee, arsenic, and sugar on the top
shelf; snuff, tea, and salt on the bottom shelf)...
- Where is the Prize? [4/17/1996]
"One of these three boxes is filled with precious jewels. The other two
contain pebbles from a pond..."
- Which Twin is Telling the Truth? [3/2/1996]
At a fork in a road are identical twins. One always lies and one always
tells the truth, but you don't know which is which. If you could only ask
one question to find out which way to go, what would it be?
- Who Gets the Job? [12/11/2001]
Each job candidate can see the other two candidates' black or red dots
but not his own. Whoever can figure out the color of his own dot gets the
- Who Got Engaged to Whom? [11/27/2001]
Dorothy, Jean, Virginia, Bill, Jim, and Tom became engaged to one
another. Who got engaged to whom?
- Who is a Liar, Who Tells the Truth? [9/4/1996]
Swimmers always tell the truth, non-swimmers always lie. If you meet
three such people, and ask them... which of these people is a swimmer or
- Who Is the Youngest Boy? [04/28/2002]
Al, Ed, and Tom are different ages. One of the following statements is
true: Ed is the oldest; Al is not the oldest; Tom is not the youngest.
Who is the youngest boy?
- Who Made Which Toys? [12/21/1998]
A math logic problem, from a rhyme describing Santa's toymakers.
- Who Owns the Fish? (Einstein's Problem) [07/18/2002]
There are 5 houses sitting next to each other, each with a different
color, occupied by 5 guys from 5 different countries...
- Who Picked the Most? [09/12/1998]
Arrange the names of the people in the order of the number of peaches
that each picked, starting with the person who picked the most.
- Why Not Force Dividing by Zero? [10/22/2002]
Has there ever been an attempt to force dividing by zero just as it
has been done so successfully by forcing the square root of -1 to be
- Why p and q? [11/29/2001]
We are wondering why the letters p and q are used to abbreviate
- Why Straightedge and Compass Only? [10/02/2002]
My geometry students want to know why constructions can only be done
using a straightedge and a compass.
- Why the Motionless Runner Parodox Fails [01/01/2005]
I read about the Motionless Runner paradox on your site, and I am now
convinced that motion is an illusion. Can you help me understand why
the paradox can't be true?