Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 115418 by mohammad17 last updated on 25/Sep/20

Commented by mohammad17 last updated on 25/Sep/20

help me sir i want (H.W)

$${help}\:{me}\:{sir}\:{i}\:{want}\:\left({H}.{W}\right) \\ $$

Answered by Dwaipayan Shikari last updated on 25/Sep/20

For any Set  A∩∅=A  A∩A    =A  Suppose x∈A   So x∈A∩A  And A∩A=A  A∩C=C∩A  Suppose x∈A   and x∈C  x∈A∩C  x∈C∩A  So C∩A=A∩C

$$\mathrm{For}\:\mathrm{any}\:\mathrm{Set} \\ $$$$\mathrm{A}\cap\varnothing=\mathrm{A} \\ $$$$\mathrm{A}\cap\mathrm{A}\:\:\:\:=\mathrm{A} \\ $$$$\mathrm{Suppose}\:\mathrm{x}\in\mathrm{A}\: \\ $$$$\mathrm{So}\:\mathrm{x}\in\mathrm{A}\cap\mathrm{A} \\ $$$$\mathrm{And}\:\mathrm{A}\cap\mathrm{A}=\mathrm{A} \\ $$$$\mathrm{A}\cap\mathrm{C}=\mathrm{C}\cap\mathrm{A} \\ $$$$\mathrm{Suppose}\:\mathrm{x}\in\mathrm{A}\:\:\:\mathrm{and}\:\mathrm{x}\in\mathrm{C} \\ $$$$\mathrm{x}\in\mathrm{A}\cap\mathrm{C} \\ $$$$\mathrm{x}\in\mathrm{C}\cap\mathrm{A} \\ $$$$\mathrm{So}\:\mathrm{C}\cap\mathrm{A}=\mathrm{A}\cap\mathrm{C} \\ $$$$ \\ $$

Answered by Dwaipayan Shikari last updated on 25/Sep/20

x is an element of set A (x∈A)  And (x∈B)  And x∈( A∩B ) (By definition)  And (A∩B)⊆A   Also (A∩B)⊆B

$$\mathrm{x}\:\mathrm{is}\:\mathrm{an}\:\mathrm{element}\:\mathrm{of}\:\mathrm{set}\:\mathrm{A}\:\left(\mathrm{x}\in\mathrm{A}\right) \\ $$$$\mathrm{And}\:\left(\mathrm{x}\in\mathrm{B}\right) \\ $$$$\mathrm{And}\:\mathrm{x}\in\left(\:\mathrm{A}\cap\mathrm{B}\:\right)\:\left(\mathrm{By}\:\mathrm{definition}\right) \\ $$$$\mathrm{And}\:\left(\mathrm{A}\cap\mathrm{B}\right)\subseteq\mathrm{A}\: \\ $$$$\mathrm{Also}\:\left(\mathrm{A}\cap\mathrm{B}\right)\subseteq\mathrm{B} \\ $$$$ \\ $$

Commented by mohammad17 last updated on 25/Sep/20

yes sir i want other question in same this method

$${yes}\:{sir}\:{i}\:{want}\:{other}\:{question}\:{in}\:{same}\:{this}\:{method} \\ $$

Commented by Dwaipayan Shikari last updated on 25/Sep/20

It is better to use Venn diagram in Q7

$$\mathrm{It}\:\mathrm{is}\:\mathrm{better}\:\mathrm{to}\:\mathrm{use}\:\mathrm{Venn}\:\mathrm{diagram}\:\mathrm{in}\:\mathrm{Q7} \\ $$

Answered by MWSuSon last updated on 25/Sep/20

by using the definition of set equality  and logical equivalence  X=Y iff  ∀x∈U, (x∈X⇔x∈Y) we can  show 6 and 7.  6)let x∈(A∩(B∪C))⇔x∈A∧x∈(B∪C)                            ⇔x∈A∧(x∈B∨x∈C)                      ⇔(x∈A∧x∈B)∨(x∈A∧x∈C)                        ⇔x∈(A∩B)∪(A∩C)  therefore A∩(B∪C)=(A∩B)∪(A∩C)  same way for 7

$$\mathrm{by}\:\mathrm{using}\:\mathrm{the}\:\mathrm{definition}\:\mathrm{of}\:\mathrm{set}\:\mathrm{equality} \\ $$$$\mathrm{and}\:\mathrm{logical}\:\mathrm{equivalence} \\ $$$$\mathrm{X}=\mathrm{Y}\:\mathrm{iff}\:\:\forall\mathrm{x}\in\mathrm{U},\:\left(\mathrm{x}\in\mathrm{X}\Leftrightarrow\mathrm{x}\in\mathrm{Y}\right)\:\mathrm{we}\:\mathrm{can} \\ $$$$\mathrm{show}\:\mathrm{6}\:\mathrm{and}\:\mathrm{7}. \\ $$$$\left.\mathrm{6}\right)\mathrm{let}\:\mathrm{x}\in\left(\mathrm{A}\cap\left(\mathrm{B}\cup\mathrm{C}\right)\right)\Leftrightarrow\mathrm{x}\in\mathrm{A}\wedge\mathrm{x}\in\left(\mathrm{B}\cup\mathrm{C}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Leftrightarrow\mathrm{x}\in\mathrm{A}\wedge\left(\mathrm{x}\in\mathrm{B}\vee\mathrm{x}\in\mathrm{C}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Leftrightarrow\left(\mathrm{x}\in\mathrm{A}\wedge\mathrm{x}\in\mathrm{B}\right)\vee\left(\mathrm{x}\in\mathrm{A}\wedge\mathrm{x}\in\mathrm{C}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Leftrightarrow\mathrm{x}\in\left(\mathrm{A}\cap\mathrm{B}\right)\cup\left(\mathrm{A}\cap\mathrm{C}\right) \\ $$$$\mathrm{therefore}\:\mathrm{A}\cap\left(\mathrm{B}\cup\mathrm{C}\right)=\left(\mathrm{A}\cap\mathrm{B}\right)\cup\left(\mathrm{A}\cap\mathrm{C}\right) \\ $$$$\mathrm{same}\:\mathrm{way}\:\mathrm{for}\:\mathrm{7} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com