Question

without using Venn diagram, prove A-(B intersect C) = (A- B) union ( A- C)​

Answer 1

Answer:

Use a Venn - Gauss diagram · Show, alternatively (which is actually exactly the same method as 1 above) that every member of the set on the left is a member of the set defined on the ... · या

Answer 2

Answer:

plz mark as brainliest

Step-by-step explanation:

Assume  a∈(A∖(B∩C))  and then derive that  a∈((A∖B)∪(A∖C)) . Using conditional introduction and then universal introduction, you can derive  ∀x(x∈(A∖(B∩C))→x∈((A∖B)∪(A∖C)) .

Assume  a∈((A∖B)∪(A∖C))  and derive that  a∈(A∖(B∩C)) . Using conditional introduction and then universal introduction, you can derive  ∀x(x∈((A∖B)∪(A∖C))→x∈((A∖B)∪(A∖C)) .

If you can show both of those conditional statements then by the axiom of extensionality the two sets are equal.To prove this you will need to recall what the following mean in terms of set membership (along with DeMorgan’s laws and the distributive property):

Set intersection:  A∩B={x:x∈A  and  x∈B}  

Set union:  A∪B={x:x∈A  or  x∈B}  

Set difference:  A∖B={x:x∈A  and  x∉B}  

To show (1) above, let  a∈(A∖(B∩C)) . By the definition of set difference  a∈A  and  a∉(B∩C) . By the definition of set intersection,  a∈A  and not  (a∈B  and  a∈C) . By DeMorgan’s law,  a∈A  and ( a∉B  or  a∉C) . By the distributive law, ( a∈A  and  a∉B ) or ( a∈A  and  a∉C ). By set difference,  a∈(A∖B)  or  a∈(A∖C) . By set union,  a∈((A∖B)∪(A∖C)) . By conditional introduction,  a∈(A∖(B∩C))→a∈((A∖B)∪(A∖C)) . By universal introduction,  ∀x(x∈(A∖(B∩C))→x∈((A∖B)∪(A∖C)) .

You can show (2) using a similar approach. and then using biconditional introduction to combine both of those subproofs together. Using the axiom of extensionality, you have shown that the sets are equal.