What is the converse of:

Question

If a=b and b=c, then a=c

If a≠c then a≠b or b≠c

a=b and b=c if, and only if, a=c

If a≠b or b≠c, then a≠c

If a=c, then a=b and b=c

Answer 1

The converse of a statement is formed by reversing the hypothesis and the conclusion. Let’s analyze the statements you've provided and find their converses:

1. Original: If \(a = b\) and \(b = c\), then \(a = c\).
   Converse: If \(a = c\), then \(a = b\) and \(b = c\).

2. Original: If \(a \neq c\), then \(a \neq b\) or \(b \neq c\).
   Converse: If \(a \neq b\) or \(b \neq c\), then \(a \neq c\).

3. Original: \(a = b\) and \(b = c\) if, and only if, \(a = c\).
   Converse: \(a = c\) if, and only if, \(a = b\) and \(b = c\). (This is still an equivalence statement)

4. Original: If \(a \neq b\) or \(b \neq c\), then \(a \neq c\).
   Converse: If \(a \neq c\), then \(a \neq b\) or \(b \neq c\).

5. Original: If \(a = c\), then \(a = b\) and \(b = c\).
   Converse: If \(a = b\) and \(b = c\), then \(a = c\).

To summarize:

1. If \(a = c\), then \(a = b\) and \(b = c\).
2. If \(a \neq b\) or \(b \neq c\), then \(a \neq c\).
3. \(a = c\) if, and only if, \(a = b\) and \(b = c\).
4. If \(a \neq c\), then \(a \neq b\) or \(b \neq c\).
5. If \(a = b\) and \(b = c\), then \(a = c\).