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Question 1196105: Without tables prove thet [(A→B)&(B→C)]→ (A→C)
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(53751)   (Show Source):
You can put this solution on YOUR website! .
If A implies B and B implies C, then A implies C.
If B follows A and C follows B, then C follows A.
It is one of axioms of human logic.
Answer by math_tutor2020(3835)  (Show Source):
You can put this solution on YOUR website!
Imagine 3 dominoes labeled A,B,C
The notation A→B can be thought of as domino A falling to knock down domino B
I.e. "A knocks down B"
or "A leads to B"
Then B→C means "B knocks down C"
The two events (A→B)&(B→C) chain together to mean: "A knocks down B" and "B knocks down C"
Rephrased: "A leads to B leads to C"
We can then cut out the middleman and conclude that knocking down A ultimately leads to C falling, which gives us A→C
This process can be extended to more than 3 terms.
For more information, check out the transitive property.
A similar related idea is the term "syllogism" found in logic class. More specifically its often called a "hypothetical syllogism"
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A real world example:
A = it rains
B = the roads get wet
C = there's a higher chance of an accident
Premise1: (A→B) = If it rains, then the roads get wet
Premise2: (B→C) = If the roads get wet, then there's a higher chance of an accident
Conclusion: (A→C) = If it rains, then there's a higher chance of an accident
Event A causes event B which in turn leads to event C.
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