(P→Q)∧(Q→R)=(P→R)

To prove the statement "(P→Q)∧(Q→R)=(P→R)", we need to show that both sides of the equation are logically equivalent.

Step 1: Assume the left-hand side (LHS) of the equation is true, which means "(P→Q)∧(Q→R)" is true.

Step 2: From the assumption, we can break down the conjunction "(P→Q)∧(Q→R)" into two separate implications: "P→Q" and "Q→R".

Step 3: Now, let's assume P is true. From the implication "P→Q", we can conclude that Q must also be true.

Step 4: Since Q is true, we can use the implication "Q→R" to conclude that R must also be true.

Step 5: Therefore, from the assumption that P is true, we have shown that R is true. This implies that "P→R" is true.

Step 6: Now, let's assume the right-hand side (RHS) of the equation is true, which means "P→R" is true.

Step 7: From the assumption, we can conclude that if P is true, then R must also be true.

Step 8: Now, let's assume P is true. From the implication "P→R", we can conclude that R must also be true.

Step 9: Since R is true, we can use the implication "Q→R" to conclude that Q must also be true.

Step 10: Therefore, from the assumption that P is true, we have shown that Q is true. This implies that "P→Q" is true.

Step 11: Combining the implications "P→Q" and "Q→R", we can conclude that "(P→Q)∧(Q→R)" is true.

Step 12: Thus, we have shown that both sides of the equation are true, which proves that "(P→Q)∧(Q→R)=(P→R)".