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Describe the important relationships between quantifiers and connectives

There are following four important relationships between quantifiers and connectives. They are used frequently in reasoning.

1. ∀x [ P(x) ⋀Q(x) ] ⇔[ ∀x P(x) ⋀∀x Q(x) ]

2. [ ∀x P(x) ⋁∀x Q(x) ] ⇒∀x [ P(x) ⋁Q(x) ]

3. ∃x [ P(x) ⋁Q(x) ] ⇔[ ∃x P(x) ⋁∃x Q(x) ]

4. ∃x [ P(x) ⋀Q(x) ] ⇒[ ∃x P(x) ⋀∃x Q(x) ]

Let us see what these mean with examples.

Let P(x) represent "x is rich", and Q(x) "x is happy", and let the universe be a set of three people. Also let LHS (RHS) denote the left (right) hand side of the implication or equivalence. Then

1. ∀x [ P(x) ⋀Q(x) ] ⇔[ ∀x P(x) ⋀∀x Q(x) ] can be illustrated as in Figure:

That is, LHS says everyone is rich and happy, and RHS says everyone is rich and everyone is happy. Thus LHS and RHS describe the same situation. That is, LHS is true if and only if RHS is true.

2. [ ∀x P(x) ⋁∀x Q(x) ] ⇒∀x [ P(x) ⋁Q(x) ] for the same interpretation as 1. above can be shown in Figure:

That is, LHS says everyone is rich or everyone is happy, and RHS says everyone is rich or happy. Thus if LHS is true, then RHS is certainly true. However on RHS it can happen that two people are rich but the third is not rich but happy. In that case LHS is not true while RHS is true. Thus RHS does not necessarily imply LHS.

3. ∃x [ P(x) ⋁Q(x) ] ⇔[ ∃x P(x) ⋁∃x Q(x) ] , again for the same example, can be shown in Figure:

LHS says someone is rich or happy, and RHS says someone is rich or someone is happy. Thus clearly LHS implies RHS. Also if someone is rich then that person is certainly rich or happy. Thus RHS implies LHS.

4. ∃x [ P(x) ⋀Q(x) ] ⇒[ ∃x P(x) ⋀∃x Q(x) ] , for the same example, can be shown in Figure:

LHS say someone is rich and happy. Hence there is someone who is rich and there is someone who is happy. Hence LHS implies RHS. However, since RHS can be true without anyone being rich and happy at the same time, RHS does not necessarily imply LHS.