Let A={1,2,3},B={3,4} and C={4,5,6}. Find
(i) A×(B∩C)
(ii) (A×B)∩(A×C)
(iii) A×(B∪C)
(iv) (A×B)∪(A×C)
Get the complete, step-by-step math solution for: "Let {A}=\{1{,}2{,}3\}, {B}=\{3{,}4\} and {C}=\{4{,}5{,}6\}. Find (i) {A} ×( {B} {C}) (ii) ( {A} × {B}) ( {A} × {C}) (iii) {A} ×( {B} {C}) (iv) ( {A} ×...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Calculate B intersection C
First, we find the intersection of sets B and C. The intersection of two sets contains all elements that are common to both sets. In this case, the only common element between B and C is 4.
Step 2: Calculate A cross (B intersection C)
Now we calculate the Cartesian product of set A with the result of the intersection, B∩C. The Cartesian product of two sets creates ordered pairs where the first element comes from the first set and the second element comes from the second set.
Step 3: Calculate A cross B and A cross C
To find (A×B)∩(A×C) and (A×B)∪(A×C), we first need to calculate the Cartesian products A×B and A×C.
Step 4: Calculate (A cross B) intersection (A cross C)
Now we find the intersection of the two Cartesian products A×B and A×C. We look for ordered pairs that are present in both sets.
Step 5: Calculate B union C
For the third part, we first find the union of sets B and C. The union of two sets contains all unique elements from both sets.
Step 6: Calculate A cross (B union C)
Finally, we calculate the Cartesian product of set A with the union of B and C. This involves forming ordered pairs with each element of A and each element of (B∪C).
Step 7: Calculate (A cross B) union (A cross C)
For the last part, we find the union of the two Cartesian products A×B and A×C. This set will contain all unique ordered pairs from both A×B and A×C.