Finding equivalent statements with quantifiers

Question

Find equivalent pairs:

a. $\forall x(P(x)\land Q(x))$

b. $(\forall x(P(x))\land (\forall xQ(x))$

c. $\exists x(P(x)\land Q(x))$

d. $(\exists x(P(x))\land (\exists x Q(x))$

Are there rules for $\forall$ and $\exists$ distribution? should I use a logic table?

Our is it for example that for $\forall x(P(x)\land Q(x))$ we first set a $x$ and then "test" the statement and therefore it is equivalent to $\exists x(P(x)\land Q(x))$?

Answer 1

You can just translate your statements in English and see if that helps.

a

For all x, both P(x) and Q(x) are true.

b

For all x, P(x) is true and for all x, Q(x) is true.

c

There exists x (or there is/are some value(s) of x) for which both P(x) and Q(x) are true.

d

There exists x for which P(x) is true and there exists x for which Q(x) is true.

Conclusion

a,b are clearly equivalent. a is clearly not equivalent to c,d. d doesn't guarantee existence of x for which P(x) and Q(x) are simultaneously true. So d is not equivalent to c. To understand this let's use the following example:

P(x)= x is black, Q(x)= x is not black. If x is a bird, d suggests some birds are black and some are not but c suggests there is one or more bird that is simultaneously black and not black.