SOLUTION: Without tables prove thet [(A→B)&(B→C)]→ (A→C)

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!
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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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