(Logic) Hi, im just struggling to understand implications

[candid nimbus]

so for p ⟶ q, why for this to be false does p need to be true and q false, can someone explain, i can't really grasp the logic for this, like in this question i understand conjunction, disjunction and symmetric difference just not the implication

[south stirrup]

Consider P = x > 5, Q = x > 10. Walk me through the 4 cases

[subtle zodiac]

make truth values first

[subtle zodiac]

wdym why P should be true and Q false?

in order to determine if it is tautology, or fallacy or what, you need to construct table of truth values

[candid nimbus]

yeah ik n i have constructed it, im just struggling grasping the logic behind the one case where p implies q is false which is when p is true n q is false

[candid nimbus]

1. p = t and q = t
   since x is greater than 5 and greater than 10 the implication is true, makes sense because "if p then q"

2. p = f and q = f
   since both are false, x is smaller than 10 and 5 the implication is true because "if p then q" both are false so it makes sense

3. p = f and q = t
   so if p is false it means x is less than 5, if q is true it means x is larger than 10, so does that mean its exactly 10 in this case? coz if thats not the case im not sure how the implication is true, is x a completely seperate variable in both instances? coz that would make more sense.

4. p = t and q = f
   so "if p then q", reason it evaluates to false is because the implication doesn't hold up, despite p being true, q is not so it makes sense its false

sry i tried getting my lecturer to explain it, they kinda explained bit said we don't need the proofs so im jus trynna understand it logically

[south stirrup]

And which cases are you uncertain about

[candid nimbus]

the one where p is false and q is true

why is it true if the implication doesn't happen

[south stirrup]

Because the implication was not strictly violated. The implication claims that whenever x is greater than 5, x must also be greater than 10.

If x = 11, P is true, Q is true and the implication holds.

If x = 6, P is true, Q is false so the implication is violated.

If x = 4, P is false and it does not matter what Q is because the implication only makes the claim for when x is greater than 5. So it defaults to true