Understanding the truth table of → (implies)

[MathWarrior]

The truth table of it is as follows for reference:

p q p→q
T T T
T F F
F T T
F F T

I was wondering anyone can shed some light on an easier way to memorize or think about this then just memorizing the truth table. This seems to be the least intuitive logical function, to understand for me.

 

[johnqwertyful]

"If I win the lottery (P), I will give you a million dollars (Q)."

When did I lie to you?

 

[Stephen Tashi]

I was wondering anyone can shed some light on an easier way to memorize or think about this then just memorizing the truth table.

It should be easy to memorize since there is only one way to make "p implies q" false.

It will be easier to understand when you study statement-functions that have a variable and are quantifed by "for each". For example " for each number x , if x > 0 then x^3 > 0". If someone says "No! That's false. Suppose x = -1", we don't consider that such an example makes the statement false. The only way to show such a quantified statement is false is to provide an example where the "if..." part is true and the "then..." part is false.

("p implies q" has the same meaning as "if p then q".)

 

[Mark44]

A slightly different way to look at it than Stephen Tashi described is this.

p q p→q
T T T
T F F
F T T
F F T

For the first two rows, the implication is true when both p and q are true, and the implication is false when p is true but q is false.
For the 3rd and 4th rows, when p is false, the implication is defined to be true, regardless of the value of q.

Using johnqwertyful's example, the only scenario that you would have a complaint about (i.e., that his implication is false) is when he actually does win the lottery (P is true), but he doesn't give you the million dollars (Q is false).

If he hasn't won the lottery, could give you the money or not give you the money, and the implication would still be considered true.

 

[blue_raver22]

I can offer some tips and strategies for understanding and memorizing the truth table for the logical function "implies."

Firstly, it is important to understand the concept behind "implies." This logical function represents a conditional statement, meaning that if the first statement (p) is true, then the second statement (q) must also be true. This is similar to the concept of cause and effect - if a certain condition is met, then a specific outcome will occur.

One way to remember the truth table is to think of it in terms of real-life examples. For instance, imagine that p represents the statement "It is raining" and q represents the statement "I will bring an umbrella." The truth table shows that if p is true (it is raining), then q must also be true (I will bring an umbrella). This can help to solidify the concept in your mind and make it easier to remember.

Another strategy is to break down the truth table into smaller parts. For example, you can focus on the first two rows where p is true and see that q must also be true in order for the statement to be true. Then, you can focus on the last two rows where p is false and see that q can be either true or false for the statement to still be true. This can help to simplify the table and make it easier to remember.

It may also be helpful to practice with some examples and see if you can identify the correct truth value for p→q. For instance, if p is "The movie starts at 7 PM" and q is "I will arrive on time," what is the truth value for p→q? It is true, because if the movie starts at 7 PM, then I must arrive on time.

Overall, understanding the concept behind "implies" and practicing with examples can help to make the truth table easier to remember. It may also be helpful to review and practice regularly to reinforce your understanding.