Use truth tables to determine the validity or invalidity of each of the followin

Question

Use truth tables to determine the validity or invalidity of each of the following arguments:

If the preident cuts social security benefit payments he will lose the support of the senior citizens, and if he cuts defense spending then he will lose the suppor of the conservatives. If the president loses the support of either the senior citizens or the conservatives then his influence in the senate will dimish. But his influence in the Senate will not diminsh. Therefore the president will not cut either social security benefits or defense spending.

I am having trouble with coming up with the initial expression, i just would like this statement written in propositional logic.

Explanation / Answer

Let assume propositions
A =>  President lose the support of the senior citizens.
B => President lose the support of the conservatives.

We need to represent If the president loses the support of either the senior citizens or the conservatives then his influence in the senate will dimish

=> if either A or B is true, president influence will dimish.
=> (A V B) must be false
=> ¬ (A V B) must be true
=> (¬ A ¬ B) must be true

so final expression is
(¬ A ¬ B)

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