Can someone explain me why the truth table of p-->q, given that p and q are any two propositions, does not agree with the truth values of the direct proof of the following example?
Example : Prove that if x is an even integer, then x^2 is also an even integer using direct proof :
Let a,b denote x is an even integer and x^2 is an even integer. We usually take a-->b to prove such examples using direct proof?
So the above example's truth table takes following form :
a , b , a-->b
1) T , T , T
2) T , F , F
3) F , T , F
4) F , F , T
I know that the general truth table of p-->q takes following form :
p , q , p-->q
1) T , T , T
2) T , F , F
3) F , T , T
4) F , F , T
Can someone explain what's the reason for the 3rd row of above truth tables' to be different?
Thanks!
The reason the truth table for the statement "p-->q" is different from the truth table of the direct proof example is because they are representing different things.
In the direct proof example, we are proving the implication "if x is an even integer, then x^2 is also an even integer". We denote "x is an even integer" as a, and "x^2 is an even integer" as b. This means when a is true, b must also be true for the statement to hold. In this case, the truth table is based on whether the statement is true or false for every possible combination of a and b.
On the other hand, the truth table for "p-->q" represents the logical implication between two propositions p and q. The statement "p-->q" is true if and only if p is false or q is true. In other words, for "p-->q" to be true, if p is true, then q must also be true. If p is false, it doesn't matter what q is, the implication is still considered true. This explains why the third row of the truth table for "p-->q" is different from the direct proof example, as it follows this logical definition.
In summary, the truth table for the direct proof example is specific to that particular implication being proven, while the truth table for "p-->q" represents the general logical implication between any two propositions p and q.