Review the Symmetric, Transitive, and Substitution Properties of Equality, and exchange the equals sign to different relations.
See solution.
Practice makes perfect
Let's begin by reviewing the Symmetric, Transitive, and Substitution Properties of Equality.
Symmetric Property of Equality: If a=b, then b=a.
Transitive Property of Equality: If a=b and b= c, then a= c.
Substitution Property of Equality: If a=b, then a may be replaced by b in any equation or expression.
We will think of real-world examples and then of real-world non-examples of these properties one at a time.
Symmetric
Let's think of an example of the Symmetric Property first. Then we will think about a non-example.
Example
We have to think of a relation that can replace the equals sign in the Symmetric Property of Equality. Let's think about a and b as two girls — Girl A and Girl B. We can exchange the equals sign for is a sister of. Then we get the following statement.
IfGirl A is a sister of Girl B , then
Girl B is a sister of Girl A. âś“
We can tell that it is true, so this is a good example of the Symmetric Property for all girls.
Non-Example
Now, we want to think of a relation that is not an example of the Symmetric Property. Let's think about a and b as people A and B. We can exchange the equals sign for is older than. Then we get the following statement.
IfA is older than B , then
B is older than A. *
We can tell that is it not true, so it is a good non-example of the Symmetric Property for all human beings.
Transitive
We will think about an example and about a non-example of the Transitive Property one at a time.
Example
Let's again think about a, b, and c in the Transitive Property of Equality as people A, B, and C. We can exchange the equals sign for has the same birthday as. We get the following statement.
IfA has the same birthday as B and B has the same birthday as C, then
A has the same birthday as C. âś“
We can tell that it is true, so this is a good real-world example of the Transitive Property.
Non-Example
To write a non-example of the Transitive Property, we can exchange the equals sign for is a mother of. Again, we will think about a, b, and c in the Transitive Property of Equality as about people A, B, and C. This gives us the following statement.
IfA is a mother of B and B is a mother of C, then,
A is a mother of C. *
We can tell that is it not true, as A would be a grandmother of C if the hypothesis was true. Therefore, it is a good real-world non-example of the Transitive Property.
Substitution
Finally, we will think about an example and about a non-example of the Substitution Property.
Example
As in the Transitive Property, let's do it for people A and B. We can exchange the equals sign for has the same height as. We get the following statement.
IfA has the same height as B, then A may be replaced by B in any equation or expression. âś“
This statement is true, however notice that the expression or equation has to involve the relation has the same height as. Therefore, it is a good real-world example of the Substitution Property.
Non-Example
We will think about a non-example of the Substitution Property in the real-world for any two human beings A and B. We can exchange the equals sign for has a different eye color than. We get the following statement.
IfA has a different eye color than B, then A may be replaced by B in any equation or expression. *
We can tell that it is not true, so it is a good non-example of the Substitution Property.
