# Learning Logic and Truth Tables: My Algebra Journey

• Shackleford
In summary, the last column in the table asks if the implication (p ∧ q) ⇒ p is true because (p ∧ q) gives you no information on whether p is true.
Shackleford
I'm taking Abstract Algebra right now, and we just briefly covered Logic and Truth Tables. This is my first time in school to learn such things.

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110712_195458.jpg

37. I understand.

39. I understand.

41. I don't understand the last column. Is the implication (p ∧ q) ⇒ p true because (p ∧ q) gives you no information on whether p is true? Why?

43. For p ⇒ q, I understand the first two column entries (T,F), is the implication because p being False gives no information on whether p ⇒ q is True? Again, for the last column, is
(p ∧ (p ⇒ q)) True because it being False gives you no new information on q?

If my reasoning is correct, then I can see the following logical consistencies here. Also, I notice for the implications ⇒, there is an additional column asking for its truth value. But, there is no such thing for the iff ⇔ statements. Why?

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41 is called vacuously true, see
http://en.wikipedia.org/wiki/Vacuous_truth

basically in a statement
$$P \implies Q$$

if P is false, then the statement is vacuously true, as the rest of the statement no longer needs to be evaluated

consider in the form
if P then Q
when P is false the statement has no more information. In an analogy with programming, there is no elseif or else so nothing else to evaluate making it vacuously true

41. is true because the only way for $A\implies B$ to be false is for A to be true and B to be false.

(p AND q) is always false when p is false.

Shackleford said:
...

43. For p ⇒ q, I understand the first two column entries (T,F), is the implication because p being False gives no information on whether p ⇒ q is True?

I would word this as: When p is false , then p ⇒ q gives no information as to whether q is true or false.

Shackleford said:
41. I don't understand the last column. Is the implication (p ∧ q) ⇒ p true because (p ∧ q) gives you no information on whether p is true? Why?

SammyS said:
41. is true because the only way for $A\implies B$ to be false is for A to be true and B to be false.

(p AND q) is always false when p is false.

Another way to think of this (#41) is this: If p and q are true, then obviously p is true. There's a related statement -
(p ∧ q) ⇒ q

Okay. I got it. Thanks for the help, guys.

## 1. What is logic and how is it related to algebra?

Logic is the study of reasoning and argumentation. It is closely related to algebra because both fields use symbols and rules to manipulate those symbols in a systematic way to reach a desired conclusion.

## 2. Why is learning logic and truth tables important?

Learning logic and truth tables is important because it helps to develop critical thinking skills and the ability to construct logical arguments. It also provides a foundation for advanced mathematical concepts and problem-solving.

## 3. What are truth tables and how are they used in logic?

A truth table is a tool used in logic to determine the truth value of a compound statement based on the truth values of its individual components. It lists all possible combinations of truth values for the statement's variables and shows the resulting truth value for the entire statement.

## 4. How can I improve my understanding of logic and truth tables?

Practice is key to improving your understanding of logic and truth tables. Start by familiarizing yourself with the basic rules and symbols, and then work through examples and exercises to apply your knowledge. You can also seek out additional resources and seek help from a teacher or tutor if needed.

## 5. Can logic and truth tables be applied to real-life situations?

Yes, logic and truth tables can be applied to real-life situations. They can help with problem-solving, decision-making, and evaluating arguments or claims. For example, if you are trying to decide between two options, you can use truth tables to evaluate the logical consequences of each choice.

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