Set Theory- a couple of questions im stuck with to do with cartesian products...

For (a), the approach I would take is something like:

$(a, b)\in(P\times Q)\cap(R\times S)$

$\Leftrightarrow (a,b)\in(P\times Q) \land (a,b)\in(R\times S)$

then show which sets a and b each have to be in and finish off by showing that $(a,b)\in(P\cap R)\times(Q\cap S)$. Make sure to use the double arrows in your working so that you can show that the equivilance works both ways.

(b) Here you want to show that (a,b) in the left hand set implies (a,b) is in the right hand set but your argument will only need to work one way.

I would start this question off with a similar approach to part (a), showing (a in P and b in Q) or (a in R and b in S).

Then use a distributivity law of logic which says $(w\land x)\lor(y\land z)\Leftrightarrow (w\lor y)\land(w\lor z)\land(x\lor y)\land(x\lor z)$. Use that to expand out your previous statement.

Now if x and y are statements then $x\land y\Rightarrow x$ so you can use that law to delete some of your terms in the previous statement. Try and choose terms to delete that will leave you with a statement saying something along the lines of "a is in P or R and b is in Q or S". That way, you'll just have a couple more lines of working to eventually say that (a,b) is in the set given on the right hand side of the question.

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Maybe would result in a possibly correct answer but the entirely wrong way to think about proving general propositions in mathematics.

For instance - how do you know you would be able to get an argument to containing nothing but logical equivalences at every step? Usually a single direction is trivial and the reverse is non trivial.

But most importantly you won't be able to prove anything of reasonable depth if you try to push everything in terms of logic.

Maybe would result in a possibly correct answer but the entirely wrong way to think about proving general propositions in mathematics.

For instance - how do you know you would be able to get an argument to containing nothing but logical equivalences at every step? Usually a single direction is trivial and the reverse is non trivial.

But most importantly you won't be able to prove anything of reasonable depth if you try to push everything in terms of logic.

Doesn't union mean either one or the other, but not both?

while that is most likely true, i should'v mentioned that this unit im doing is heavily based around logic and so would likely be expected to use logic in the answer here,

& one other thing i notice you mentioned...
"It follows that x is either an element of PxQ or RxS (or both)"
Doesn't union mean either one or the other, but not both?

& thanks to both of you for taking the time to help

Then use a distributivity law of logic which says $(w\land x)\lor(y\land z)\Leftrightarrow (w\lor y)\land(w\lor z)\land(x\lor y)\land(x\lor z)$. Use that to expand out your previous statement.

Now if x and y are statements then $x\land y\Rightarrow x$ so you can use that law to delete some of your terms in the previous statement. Try and choose terms to delete that will leave you with a statement saying something along the lines of "a is in P or R and b is in Q or S". That way, you'll just have a couple more lines of working to eventually say that (a,b) is in the set given on the right hand side of the question.

i don't understand this bit , i can't see how x^y => x if x and y are statements , or in this case, (aEP)^(bEQ) => (aEP)

i don't understand this bit , i can't see how x^y => x if x and y are statements , or in this case, (aEP)^(bEQ) => (aEP)