Substitution

Substitution is a conditional property that enables replacement of one input (expression) by another input that is equal. Consider the following diagram: 3 5 adder 8

This is an adder that takes inputs 3 and 5. Since

3 + 5 = 8

The output is 8.

Now suppose we are told that

a = 3

Then we can replace 3 with a, since they are equal in value: a 5 adder 8

The output remains 8. Swapping one of the inputs with an equal value did not affect the output. In other words:

a + 5 = 3 + 5

Here are more examples:

if a = b, then a + 5 = b + 5

if x = 45, then cosine of x = cosine of 45

if y = a + b, then y + z = (a + b) + z

if x = 5/y, then x ⋅ z = (5/y) ⋅ z

In Logicwalk, to perform a substitution, the condition must be an equality comparison (an expression with an equal sign).

The conclusion is also an equality comparison. Here is the process of writing the conclusion:

  1. We write some expression that uses the left side of the condition.
  2. We write an equal sign and repeat the same expression on the right side of the equal sign.
  3. We modify the right side by replacing the left side of the condition with the right side of the condition.

Let's follow this process with one of the examples above. We are given that:

a = b

First, we write some expression using a:

a + 5

Then we write an equal sign and repeat a + 5:

a + 5 = a + 5

Finally, we replace a with b:

a + 5 = b + 5

Next, let's look at an example of using substitutions in a proof.

Suppose that John has 12 dollars in his pocket, and Mike has 5 more dollars than John. How much does Mike have?

For simplicity, let's say that M is the amount of money in Mike's pocket and J is the amount of money in John's pocket.

Here is the 2 column proof:

Statement Reason
J = 12 dollars Given
M = J + 5 dollars Given
J + 5 dollars = 12 dollars + 5 dollars Substitution from the first given
12 dollars + 5 dollars = 17 dollars Fixed Value Statement
J + 5 dollars = 17 dollars Transitive Property
M = 17 dollars Transitive Property

Therefore, Mike has 17 dollars in his pocket.

Substitution is intuitive and also very powerful. If substitution was unavailable, we would need many more conditional properties to prove statements.


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