Let's break down the given mathematical statements and identify the properties they correspond to:
1. If [tex]\(a = b\)[/tex], then [tex]\(b = a\)[/tex]
- This is the Symmetric Property of Equality. It states that equality works both ways. If one quantity equals another, then the second quantity equals the first.
2. If [tex]\(a = b\)[/tex] and [tex]\(b = c\)[/tex], then [tex]\(a = c\)[/tex]
- This is the Transitive Property of Equality. It establishes that if one quantity equals a second quantity, and the second quantity equals a third quantity, then the first quantity equals the third quantity.
3. If [tex]\(a \neq b\)[/tex], then [tex]\(b \neq a\)[/tex]
- This is referred to as the Symmetric Property of Inequality. It implies that inequality works both ways. If one quantity is not equal to another, then the second quantity is not equal to the first.
4. If [tex]\(a = b\)[/tex], then [tex]\(a + c = b + c\)[/tex]
- This is the Addition Property of Equality. It states that if two quantities are equal, adding the same value to both quantities maintains the equality.
Based on these definitions, we can assign the following numbers to these properties:
1. Symmetric Property of Equality = 1
2. Transitive Property of Equality = 2
3. Symmetric Property of Inequality = 3
4. Addition Property of Equality = 4
Thus, the modern mathematical statements equivalent to the given axioms correspond to the following numbers:
1. Symmetric Property of Equality [tex]\( a = b \Leftrightarrow b = a = 1 \)[/tex]
2. Transitive Property of Equality [tex]\( a = b \text{ and } b = c \Leftrightarrow a = c = 2 \)[/tex]
3. Symmetric Property of Inequality [tex]\( a \neq b \Leftrightarrow b \neq a = 3 \)[/tex]
4. Addition Property of Equality [tex]\( a = b \Leftrightarrow a + c = b + c = 4 \)[/tex]
Therefore, the identified properties in order are:
[tex]\( (1, 2, 3, 4) \)[/tex]
