I didn’t realize the foundation is shaky until trying to understand Group Theory, Lie Algebra or Manifold now, hence have to go back to build up from beginning. Following The Bright Side of Mathematics, hopefully can forge ahead.

The outline he gives in the “Start Learning Mathematics” is

First, logic.

Terminologies are: logical statement(proposition) is the statement that is either true of false; if there is variable such as x +1 = 1, x is a variable, then it is not a logical statement or proposition, you have to give x a value then it’s a proposition.

There are logical operations such as negation, conjunction(and), disjunction(or). So A or negate A is always true if you draw a truth table, the combination of A or negate A is always true, hence called tautology.

If the truth table (all possible assignment of truth values for the logical valuables) are the same we call the two logical statements logical equivalent. For example

Another important logical operation is called conditional A –> B

And A <–>B, biconditional. A if and only if (iif).

With above knowledge, we can build up “deduction rule” to come to a new true proposition/logical statement given a true proposition/logical statement.

Need to analyze as below, but not successful, tbd