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Truth Table

Posted on April 3, 2009 | Leave a comment

Patrick Heusmann

Truth Table

The truth table is an issue in math that I had never heard of before today.  So what is a truth Table? The truth table is a mathematical table used in logic.  The tables are usually connected to Boolean algebra, boolean functions, and propositional calculus.  They use the tables to help find the functional values of logical expressions on each of their functional arguments.  Truth tables can also be used to tell whether an expression is logically valid.  Truth tables can only have to possible logical answer those being true or false sometimes written as t or F and also 0 or 1.  Besides true and false there are several other symbols I that are important to know before getting any further in truth tables.  Those being:

  = AND (logical conjunction)

= OR (logical disjunction)

= XOR (exclusive or)

= XNOR (exclusive nor)

= conditional “if-then”

= conditional “(then)-if”

biconditional or “if-and-only-if” is logically equivalent to : XNOR (exclusive nor).

These symbols are important to understand and recognize while using one of the many different types of truth tables.  I’m going to talk eight different types of truth tables.  The first being:

  • 1. Logical negation– is an operation on one logical value that produces a value of true if its operand is false and a value of false if its operand is true.
  • 2. Logical conjunction -is an operation on two logical values, it has a value of false if both operand are true.
  • 3. Logical disjunction– is an operation on two logical values it has a false value if both operand are false.
  • 4. Logical implication– deals with to logical values and is false if the first operand is true and the second one is false.
  • 5. Logical equality (also known as biconditional) – is an operation on two logical values has true value if both of the operand are either true or both are false.
  • 6. Exclusive disjunction -is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if one but not both of its operands is true.
  • 7. The logical NAND– is an operation on two logical values is false if both open operands are true or true if any of the operand is false.
  • 8. The logical NOR– is an operation on two logical values does the opposite thing as the logical NAND, meaning it is true if both operand are false or false if any operand are true.

Truth tables are used a lot in finding if a expression is logically true or false.  I feel that to truly understand a truth table you need an example so I am going to put the most simple one a Logical negation truth table.

Logical Negation

p

 

T

F

F

T

 

Truth tables are an will always be an important mathematical tool used to help solve and find the logical answer to expressions.

  • Enderton, H. (2001). A Mathematical Introduction to Logic, second edition, Harcourt Academic Press. ISBN 0-12-238452-0
  • Quine, W.V. (1982), Methods of Logic, 4th edition, Harvard University Press, Cambridge, MA.

 

 

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Math that is Interesting

Posted on February 12, 2009 | Leave a comment

To me calculus can be a subject that at times is extremely boring and also extremely difficult. There are things such as theorems and equation that just go straight over my head, and it’s not always that I wouldn’t be able to understand them if I tried. In most cases it’s the fact I just find them uninteresting, I feel like these things will never be used by me again after this class so I normally just don’t care about them. And probably to me the one subject in calculus that really just goes right over my head is derivatives, I know there importance to the class, but I really just can’t really find it in me to find them interesting. Although in my first few sentences it sounds like I can’t stand calculus at all there are some parts of it that I never thought would interest me before I learned about them this semester. The one thing this semester in calculus that I became very interested in when we learned about it was velocity problems, and the average velocity and instantaneous velocity that come with them. Before I get into why they interest me ill first explain what velocity problems are. We use velocity problems to find the average velocity of an object, and then use the average velocity and certain time periods to find the average of the instantaneous velocity at a certain time of the objects movement. To find the average velocity we first need to find the change in position and then divide it by the change in time. To find the instantaneous velocity we almost do the same thing except we continue to plug numbers in closer and closer to the time we want to find the instantaneous velocity at. Once you have plugged in numbers extremely close to your chosen time you are able to estimate the velocity at that time. An example of a velocity problem would be:

  • 1. If a rock is thrown upward on mars with a velocity of 10 m/s, its height t seconds later is given by y= 10t- 1.86t squared.

Avg. Velocity (1,1.1)= y(2)-Y(1)/1.1-1=6.094m/s

Avg. Velocity (1,1.01)= y(2)-y(1)/1.01-1=6.2619 m/s

Avg. Velocity (1,1.001)= y(2)-Y(1)/1.001-1=6.278 m/s

B. So if we’re are trying to find the instantaneous velocity at t= 1 we are able to estimate using the average velocity that it would be 6.28 m/s

I feel that this subject was interesting to me because it was the one thing I felt I could maybe use later on in life. I also found it interesting that you are able to find the speed of an object by using calculus.

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