List all the elements of A * B where A = {a,b,c} and B = {1,2}.?

1) List all the elements of A * B where A = {a,b,c} and B = {1,2}.





2)Given the sets X ={24k +7 : k is an element of Z} , Y = {4n+3 : n is an element of Z}, V={6m+1 : m is an element of Z} , prove that X "is a subset of" Y and X "is a subset of" V but Y "is NOT a subset of" V .





3)If A, B, C are sets such that A "is a subset of" B and B "is a subset of" C , prove that A "is a subset of" C.





4) Is it true that if P(A) = P(B) for two sets A, B then A = B ?





Please give a reason and conclusion for each of the above as I am finding it very difficult to understand my teacher's silly notes!! thank you!!!

List all the elements of A * B where A = {a,b,c} and B = {1,2}.?
1) AXB is the set of ordered pairs (x, y) such that x is in A and y is in B.


AXB = {(a,1), (a,2), (b,1), (b,2), (c,1), (c,2)}





2)


X "is a subset of" Y : let a be an element of x.


Then a = 24K+7 = 24k + 4 + 3 = 4(6k + 1) + 3


Therefore a is an element of Y.





X "is a subset of" V : let b be an element of x.


Then b = 24K+7 = 24k + 6 + 1 = 6(4k + 1) + 1


Therefore b is an element of V.





Y "is NOT a subset of" V: 3 is an element of Y since 3 = 4(0)+3 but 3 is not an element of V because if you solve for m in 3=6m+1, you get m= 1/3 which is not Z.





3) You need to show that if a is in A, then it is in C too.


Choose a in A, then it must be in B as well.


Since B "is a subset of" C, then a also has to be in C. This proves that A "is a subset of" C.





4) I think it is true.

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