2. PRELIMINARIES
We recall first the elementary notion of complex intuitionistic fuzzy sets and complex intuitionistic fuzzy subrings which play a key role for our further analysis.
Definition 2.1.
[1] A fuzzy set λ of a nonempty set P is a mapping, λ:P→[0,1].
Definition 2.2.
[4] An intuitionistic fuzzy set A of a universe of discourse P is a triplet of the form A={(m,λA(m),λ^A(m)):m∈P}, where the function λA(m):P→[0,1] and λ^A(m):P→[0,1] are represents the membership and nonmembership of an element m of P, respectively. These function must fulfill the condition 0≤λA(m)+λ^A(m)≤1.
Definition 2.3.
[9] Let A and B be any two intuitionistic fuzzy sets of the universe of discourse P. Then the following operation are defined as:
(A∩B)(m)={(m,λA∩B(m),λ^A∩B(m))|m∈P}, where λA∩B(m)=min{λA(m),λB(m)} and λ^A∩B(m)=max{λ^A(m),λ^B(m)},
(A∪B)(m)={(m,λA∪B(m),λ^A∪B(m))|m∈P}, where λA∪B(m)=max{λA(m),λB(m)} and λ^A∪B(m)=min{λ^A(m),λ^B(m)},
(A+B)(m)=(m,λA(m)+λB(m)−λA(m).λB(m),λ^A(m).λ^B(m))|m∈P,
(A.B)(m)=(m,λA(m).λB(m),λ^A(m)+λ^B(m)−λ^A(m).λ^B(m))|m∈P.
Definition 2.4.
[15] An intuitionistic fuzzy set A of a ring S is called a intuitionistic fuzzy subring S, if the following condition hold:
λA(m−n)≥min{λA(m),λA(n)},
λA(mn)≥min{λA(m),λA(n)},
λ^A(m−n)≤max{λ^A(m),λ^A(n)},
λ^A(mn)≤max{λ^A(m),λ^A(n)}, for all m,n∈S.
Definition 2.5.
[13] A complex fuzzy set A of universe of discourse P is identify with the membership function θA(m)=ηA(m)eiφA(m) and is defined as θA:P→{z∈C:|z|≤1}, where C is set of complex numbers. This membership function receive all membership value from the unit disc on complex plane, where i=−1, where both ηA(m) and φA(m) are the real valued such that ηA(m)∈[0,1] and φA(m)∈[0,2π]. In this paper, we use CFS for complex fuzzy subset.
Definition 2.7.
[19] A complex intuitionistic fuzzy set A of crisp nonempty set P is an object of the form A={(m,θA(m),θ^A(m)):m∈P} where the membership function θA(m)=ηA(m)eiφA(m) and nonmembership function θ^A(m)=η^A(m)eiφ^A(m) is defined as θA:P→{z∈C:|z|≤1} and θ^A:P→{z∈C:|z|≤1}, where C is set of complex numbers. These membership and nonmembership function receive all degree of membership and nonmembership from the unit disc on complex plane respectively such that the sum of membership and nonmembership value is also lies within unit disc of complex plane, where i=−1,ηA(m),η^A(m),φA(m) and φ^A(m) are real-valued function such that 0≤ηA(m)+η^A(m)≤1 and 0≤φA(m)+φ^A(m)≤2π. In this paper we use the complex intuitionistic fuzzy set (CIFS) for the complex fuzzy subset. For the sake of simplicity we shall use θA(m)=ηA(m)eiφA(m),θB(m)=ηB(m)eiφB(m) as the membership function and θ^A(m)=η^A(m)eiφ^A(m),θ^B(m)=η^B(m)eiφ^B(m) as the nonmembership function of CIFS A and B, respectively.
Definition 2.9.
[27]
A complex intuitionistic fuzzy set (CIFS) A is homogeneous CIFS, if for all m,x∈S, we have
ηA(m)≤ηA(x) if and only if φA(m)≤φA(x),
η^A(m)≤η^A(x) if and only if φ^A(m)≤φ^A(x).
A CIFS A is homogeneous CIFS with B, if for all m,x ∈S, we have
ηA(m)≤ηB(x) if and only if φA(m)≤φB(x),
η^A(m)≤η^B(x) if and only if φ^A(m)≤φ^B(x).
Throughout this paper we use CIFS as homogeneous CIFS.
Definition 2.10.
[27] Let A={(m,ψA(m),ψ^A(m))} be a intuitionistic fuzzy subset of S. Then the π-intuitionistic fuzzy subset Aπ is defined as
Aπ={(m,ψAπ(m),ψ^Aπ(m)):m∈S},
where
ψA(m)=2πψA(m) and
ψ^A(m)=2πψ^A(m) be membership function and non-membership function, respectively for all
m∈S. For convenience, we shall write
π-IFS for
π-intuitionistic fuzzy subset.
Definition 2.11.
[27] A π-IFS Aπ of ring S is called π-fuzzy subring (π−FSR) of S, for all m,n∈S, if
ψAπ(m−n)≥min{ψAπ(m),ψAπ(n)},
ψAπ(mn)≥min{ψAπ(m),ψAπ(n)},
ψ^Aπ(m−n)≤max{ψ^Aπ(m),ψ^Aπ(n)},
ψ^Aπ(mn)≤max{ψ^Aπ(m),ψ^Aπ(n)}.
In this paper, we write π-intutionistic fuzzy subring as π-IFSR.
Theorem 2.12.
[27] A π-IFS Aπ of ring S is a π-IFSR if and only if A is Intuitionistic fuzzy subrings (IFSR).
Definition 2.13.
[27] A CIFS A=(θA(m),θ^A(m)) of ring S is called a complex intuitionistic fuzzy subring, for all m,n∈S, if
θA(m−n)≥min{θA(m),θA(n)},
θA(mn)≥min{θA(m),θA(n)},
θ^A(m−n)≤max{θ^A(m),θ^A(n)},
θ^A(mn)≤max{θ^A(m),θ^A(n)}.
For convenience, we shall use complex intuitionistic fuzzy subrings (CIFSR) for complex intuitionistic fuzzy subring.
Definition 2.14.
[27] Let A be a CIFS of S. For r∈[0,1] and t∈[0,2π] the level subset of CIFS is defined by
A(r,t)(r^,t^)(m)={m∈P:ηA(m)≥r,φA(m)≥t,η^A(m)≤r^,φ^A≤t^}.
For t=0, we obtain the level subset Arr^={m∈P:ηA(m)≥r,η^A(m)≤r^} and for r=0, we obtain the level subset Att^={m∈P:φA(m)≥t,φ^A≤t^}.
Definition 2.15.
[27] Let f:S→R be a ring homomorphism. Let A and B be two CIFSR of ring S and R, respectively, for all x∈S, m∈R. Then the sets E={(m,f(ηA)(m),f(η^A))(m)} and F={(x,f−1(θB)(x),f−1(θ^B)(x))} are called image of A and preimage of B, respectively, where
f(θA)(m)=sup{θA(x),iff(x)=m},f−1≠ϕ0,otherwise
f(θ^A)(m)=inf{θ^A(x),iff(x)=m},f−1≠ϕ1,otherwise
f−1(θB)(x)=θB(f(x),
f−1(θ^B)(x)=θ^B(f(x).
3. PROPERTIES OF THE DIRECT PRODUCT OF COMPLEX INTUITIONISTIC FUZZY SUBRINGS
In this section, we use the concept of CIFS to define direct product of CIFS. We prove that direct product of two CIFSR is CIFSR and investigate their properties.
Definition 3.1.
Let A and B be any two π-intuitionistic fuzzy sets of sets S, for all m∈S. Then the following operation are defined as:
(Aπ∩Bπ)(m)={(m,ψAπ∩Bπ(m),ψ^Aπ∩Bπ(m))}, where ψAπ∩Bπ(m)=min{ψAπ(m),ψBπ(m)} and ψ^Aπ∩Bπ(m)=max{ψ^Aπ(m),ψ^Bπ(m)}.
(Aπ∪Bπ)(m)={(m,ψAπ∪Bπ(m),ψ^Aπ∪Bπ(m))}, where ψAπ∪Bπ(m)=max{ψAπ(m),ψBπ(m)} and ψ^Aπ∪Bπ(m)=min{ψ^Aπ(m),ψ^Bπ(m)}.
Definition 3.2.
Let A and B be two π-intuitionistic fuzzy sets of sets S1 and S1, respectively. The Cartesian product of π-intuitionistic fuzzy sets A and B is defined as
(Aπ×Bπ)(m,n)={((m×n),ψAπ×Bπ(m,n),ψ^Aπ×Bπ(m,n))}
for all
m∈S1, n∈S2.
Definition 3.4.
[20] Let A and B be two CIFS of sets S1 and S1. The Cartesian product of CIFS A and B is defined by a function
A×B={((m,n),θA×B(m,n),θ^A×B(m,n))},θA×B(m,n)=ηA×B(m,n)eiφA×B(m,n)=min{ηA(m),ηB(n)}eimin{φA(m),φB(n)},θ^A×B(m,n)=η^A×B(m,n)eiφ^A×B(m,n)=max{η^A(m),η^B(n)}eimax{φ^A(m),φ^B(n)}.
For sake of simplicity, throughout this paper we shall use θA×B(m,n)=ηA×B(m,n)eiφA×B(m,n) and θC×D(m,n)=ηC×D(m,n)eiφC×D(m,n) for the membership function of Cartesian product of CIFS A×B and C×D, respectively, and θ^A×B(m,n)=η^A×B(m,n)eiφ^A×B(m,n) and θ^C×D(m,n)=η^C×D(m,n)eiφ^C×D(m,n), for the nonmembership function of Cartesian product of CIFS A×B and C×D, respectively.
Theorem 3.5.
Let A and B be to CIFSR of S1 and S2, respectively. Then A×B is CIFSR of S1×S2.
Proof.
Let m,x∈S1 and n,y∈S2 be an elements. Then (m,n),(x,y)∈S1×S2. Consider
θA×B((m,n)−(x,y))=θA×B(m−x,n−y)=ηA×B(m−x,n−y)eiϕA×B(m−x,n−y)=min{ηA(m−x),ηB(n−y)}eimin{ϕA(m−x),ϕB(n−y)}=min{ηA(m−x)eiϕA(m−x),ηB(n−y)eiϕB(n−y)}=min{θA(m−x),θB(n−y)}≥min{min{θA(m),θA(x)},min{θB(n),θB(y)}}=min{min{θA(m),θB(n)},min{θA(x),θB(y)}}θA×B((m,n)−(x,y))≥min{θA×B(m,n),θA×B(x,y)}.
Further,θA×B((m,n)(x,y))=θA×B(mx,ny)=ηA×B(mx,ny)eiϕA×B(mx,ny)=min{ηA(mx),ηB(ny)}eimin{ϕA(mx),ϕB(ny)}=min{ηA(mx)eiϕA(mx),ηB(ny)eiϕA(ny)}=min{θA(mx),θB(ny)}≥min{min{θA(m),θA(x)},min{θB(n),θB(y)}}Consequently,θA×B((m,n)(x,y))≥min{θA×B(m,n),θA×B(x,y)}.
Moreover,θ^A×B((m,n)−(x,y))=θ^A×B((m−x),(n−y))=η^A×B(m−x,n−y)eiϕ^A×B(m−x,n−y)=max{η^A(m−x),η^B(n−y)}eimax{ϕ^A(m−x),ϕ^B(n−y)}=max{η^A(m−x)eiϕ^A(m−x),η^B(n−y)eiϕ^A(n−y)}=max{θ^A(m−x),θ^B(n−y)}≤max{max{θ^A(m),θ^A(x)},max{θ^B(n),θ^B(y)}}=max{max{θ^A(m),θ^B(n)},max{θ^A(x),θ^B(y)}}.
As a result, θ^A×B((m,n)−(x,y))≤max{θ^A×B(m,n),θ^A×B(x,y)}.
Further,θ^A×B((m,n)(x,y))=θ^A×B(mx,ny)=η^A×B(mx,ny)eiϕ^A×B(mx,ny)=max{η^A(mx),η^B(ny)}eimax{ϕ^A(mx),ϕ^B(ny)}=max{η^A(mx)eiϕ^A(mx),η^B(ny)eiϕ^A(ny)}=max{θ^A(mx),θ^B(ny)}≤max{max{θ^A(m),θ^A(x)},max{θ^B(n),θ^B(y)}}=max{max{θ^A(m),θ^B(n),max{θ^A(x)},θ^B(y)}}θ^A×B((m,n)(x,y))≤max{θ^A×B(m,n),θ^A×B(x,y)}.
Thus concluded the proof.
Corollary 3.6.
Let A1,A2…An be CIFSR of S1,S2,…Sn, respectively. Then A1×A2×…×An is CIFSR of S1×S2×…×Sn.
Example 3.8.
Let Z¯2=0,1 and S=e,a,b,c be two rings. Here S is ring of 2×2 matrices over Z¯2 with second row has zero entries, where e=0000,a=0100, b=1000,c=1100.
Z¯2×S={0,e,0,a,0,b,0,c,1,e,1,a,1,b,1,c}. Then two CIFS A and B is defined by
θA=0,0.2eiπ∕12,1,0.1eiπ∕15,θ^A=0,0.55eiπ∕6,1,0.7eiπ∕3,θB=e,0.3eiπ∕3,a,0.45eiπ∕2,(b,0.33eiπ∕3),(c,0.4eiπ),θ^B=e,0.24eiπ∕8,a,0.2eiπ∕10,(b,0.1eiπ∕9),(c,0.23eiπ∕6).
Therefore,
θA×B(x)=0.2eiπ∕12,forallm∈0,e,0,a,0,b,0,c0.1eiπ∕15,forallm∈1,e,1,a,1,b,1,cθ^A×B(x)=0.55eiπ∕6,forallx∈0,e,0,a,0,b,0,c0.7eiπ∕3,forallx∈1,e,1,a,1,b,1,c
Here, A×B is CIFSR of S1×S2 and A is CIFSR of B. But B is not a CIFSR of S2 because (B)0.4,π2.23,π6=a,c is not a subring.
Theorem 3.10.
Let A and B be two CIFS of rings S1 and S2, respectively. If A×B is a CIFSR of S1×S2, then at least one of the following assertions must be hold.
ηA0≥ηBn, φA0≥φBn, η^A(0)≤η^Bn and φ^A0≤φ^Bn,foralln∈S2,
ηB(0′)≥ηAm, φB0′≥φAm,η^B(0′)≤η^Am and φ^B0′≤φ^Am,forallm∈S1.
where
0 and
0′ are identities of
S1 and
S2, respectively.
Proof.
Let A×B be a CIFSR of S1×S2. On contrary, suppose that the assertions (1) and (2) do not hold. Then there exist m∈S1 and n∈S2 such that
ηA0≤ηBn, φA0≤φBn, η^A(0)≥η^Bn and φ^A0≥φ^Bn,foralln∈S2
ηB(0′)≤ηAm, φB0′≤φAm,η^B(0′)≥η^Am and φ^B0′≥φ^Am,forallm∈S1
Consider,θA×B(m,n)=min{ηA(m),ηB(n)}eimin{φA(m),φB(n)}≥min{ηA(0),ηB(0′)}eimin{φA(0),φB(0′)}θA×B(m,n)≥θA×B(0,0′)And,θ^A×B(m,n)=max{ηA(m),η^B(n)}eimax{φ^A(m),φB(n)}≤max{η^A(0),η^B(0′)}eimax{φ^A(0),φ^B(0′)}θ^A×B(m,n)≤θ^A×B(0,0′).
But A×B is CIFSR. Hence, at least one of the following assertions must be hold:
ηA0≥ηBn, φA0≥φBn, η^A(0)≤η^Bn and φ^A0≤φ^Bn,foralln∈S2
ηB(0′)≥ηAm, φB0′≥φAm,η^B(0′)≤η^Am and φ^B0′≤φ^Am,forallm∈S1
Theorem 3.11.
Let A and B CIFS of S1 and S2 such that ηB(0′)≥ηA(m), φB0′≥φAm,η^B(0′)≤η^A(m) and φ^B(0′)≤φ^A(m) for all m∈S1 and 0′ is identity of S2. If A×B is CIFSR of S1×S2, then A is CIFSR of S1.
Proof.
Let m,0′,(x,0′) be elements of S1×S2. By given condition ηB(0′)≥ηA(m) and φB(0′)≥φA(m), η^B(0′)≤η^A(m) and φ^B(0′)≤φ^A(m) for all m,x∈S1 and 0′∈S2.
ConsiderθA(m−x) =ηA(m−x)eiφA(m−x) =minηA(m−x)eiφA(m−x),ηB(0′−0′)eiφB(0′−0′) ={ηA×B((m,0′)−(x,0′))}ei{φA×B((m,0′)−(x,0′))} ≥min{ηA×B(m,0′),ηA×B(x,0′)} eimin{φA×B(m,0′),φA×B(x,0′)} =minmin{ηA(m),ηB(0′)},min{ηA(x),ηB(0′)} ×eimin{min{φA(m),φB(0′)},min{φA(x),φB(0′)}} =minmin{ηA(m),ηA(m)},min{ηA(x),ηA(x)} ×eimin{min{ηA(m),ηA(m)},min{ηA(x),ηA(x)}} =min{ηA(m),ηA(x)}eimin{φA(m),φA(x)} =min{θA(m),θA(x)}.Thus,θA(m−x)≥min{θA(m),θA(x)}.
Moreover,θAmx=ηA(mx)eiφA(mx)=minηA(mx)eiφA(mx),ηB(0′0′)eiφB(0′0′)={ηA×B((m,0′)(x,0′))}ei{φA×B((m,0′)(x,0′))}≥min{ηA×B(m,0′),ηA×B(x,0′)}eimin{φA×B(m,0′),φA×B(x,0′)}=minmin{ηA(m),ηB(0′)},min{ηA(x),ηB(0′)}×eimin{min{φA(m),φB(0′)},min{φA(x),φB(0′)}}=min{min{ηA(m),ηA(m)},min{ηA(x),ηA(x)}}×eimin{min{ηA(m),ηA(m)},min{ηA(x),ηA(x)}}=min{ηA(m),ηA(x)}eimin{φA(m),φA(x)}=min{θA(m),θA(x)}.Thus,θA(mx)≥min{θA(m),θA(x)}
On the other hand, we have
θ^A(m−x)=η^A(m−x)eiφ^A(m−x) =maxη^A(m−x)eiφ^A(m−x),η^B(0′−0′)eiφ^B(0′−0′) ={η^A×B((m,0′)−(x,0′))}ei{φ^A×B((m,0′)−(x,0′))} ≤max{η^A×B(m,0′),η^A×B(x,0′)} eimax{φ^A×B(m,0′),φ^A×B(x,0′)} =maxmax{η^A(m),η^B(0′)},max{η^A(x),η^B(0′)} ×eimax{max{φ^A(m),φ^B(0′)},max{φ^A(x),φ^B(0′)}} =maxmax{η^A(m),η^A(m)},max{η^A(x),η^A(x)} ×eimax{max{η^A(m),η^A(m)},max{η^A(x),η^A(x)}} =max{η^A(m),η^A(x)}eimax{φ^A(m),φ^A(x)} =max{θ^A(m),θ^A(x)}.As a result,θ^A(m−x)≤max{θ^A(m),θA(x)}θ^A(mx)=η^A(mx)eiφ^A(mx) =maxη^A(mx)eiφ^A(mx),η^B(0′0′)eiφ^B(0′0′) ={η^A×B((m,0′)(x,0′))}ei{φ^A×B((m,0′)(x,0′))} ≤max{η^A×B(m,0′),η^A×B(x,0′)} eimax{φ^A×B(m,0′),φ^A×B(x,0′)} =maxmax{η^A(m),η^B(0′)},max{η^A(x),η^B(0′)} ×eimax{max{φ^A(m),φ^B(0′)},max{φ^A(x),φ^B(0′)}} =maxmax{η^A(m),η^A(m)},max{η^A(x),η^A(x)} ×eimax{max{η^A(m),η^A(m)},max{η^A(x),η^A(x)}} =max{η^A(m),η^A(x)}eimax{φ^A(m),φ^A(x)} =max{θ^A(m),θ^A(x)}.
Consequently, θ^A(mx)≤max{θ^A(m),θA(x)}. Hence, proved our claim.
Theorem 3.12.
Let A and B two CIFS of S1 and S2 such that ηA(0)≥ηB(n), φA0≥φBn,η^A(0)≤η^B(n) and φ^A(0)≤φ^B(n) for all n∈S2 and e is identity of S1. If A×B is CIFSR of S1×S2, then B is CIFSR of S2.
Proof.
Similar as Theorem 3.11.
Corollary 3.13.
Let A and B two CIFS of S1 and S2, respectively. If A×B is CIFSR of S1×S2, then A is a CIFSR of S1 or B is a CIFSR of S2.
Theorem 3.14.
Let A×B be a CIFS of ring S1×S2. Then A×B is a CIFSR of S1×S2 if and only if:
The fuzzy set A×B¯=m,n,ηA×Bm,n,η^A×Bm,n:m,n∈S1×S2,ηA×Bm,n,η^A×Bm,n∈[0,1] is an IFSR.
The π-fuzzy set A×B̲={m,n,φA×Bm,n,φ^A×Bm,n,:m,n∈S1×S2,φA×Bm,n,φ^A×B m,n∈[0,2π]} is a π-IFSR.
Proof.
Suppose that A×B is a CIFSR and m,n,(x,y)∈S1×S2. Then we have
ηA×B(m,n−(x,y))eiφA×B((m,n)−(x,y)) =θA×B((m,n)−(x,y)) ≥min{θA×B(m,n),θA×B(x,y)} =min{ηA×B(m,n)eiφA×B(m,n),ηA×B(x,y)eiφA×B(x,y)} =min{ηA×B(m,n),ηA×B(x,y)}eimin{φA×B(m,n),φA×B(x,y)}(AsA×Bis homogeneous)ηA×B((m,n)−(x,y))≥min{ηA×B(m,n),ηA×B(x,y)},φA×B((m,n)−(x,y))≥min{φA×B(m,n),φA×B(x,y)}and,ηA×B((m,n)(x,y))eiφA×B((m,n)(x,y)) =θA×B((m,n)(x,y)) ≥min{θA×B(m,n),θA×B(x,y)} =min{ηA×B(m,n)eiφA×B(m,n),ηA×B(x,y)eiφA×B(x,y)} =min{ηA×B(m,n),ηA×B(x,y)}eimin{φA×B(m,n),φA×B(x,y)}(By using the homogeneous property)ηA×B((m,n)(x,y))≥min{ηA×B(m,n),ηA×B(x,y)},and,φA×B((m,n)(x,y))≥min{φA×B(m,n),φA×B(x,y)}Further,η^A×B((m,n)−(x,y))eiφ^A×B((m,n)−(x,y)) =θ^A×B((m,n)−(x,y)) ≤max{θ^A×B(m,n),θ^A×B(x,y)} =max{η^A×B(m,n)eiφ^A×B(m,n),η^A×B(x,y)eiφ^A×B(x,y)} =maxη^A×B(m,n),η^A×B(x,y)eimax{φ^A×B(m,n),φ^A×B(x,y)}(AsA×Bis homogeneous)η^A×B((m,n)−(x,y))≤max{η^A×B(m,n),η^A×B(x,y)}φ^A×B((m,n)−(x,y))≤max{φ^A×B(m,n),φ^A×B(x,y)}
Moreover,η^A×B((m,n)(x,y))eiφ^A×B((m,n)(x,y)) =θ^A×B((m,n)(x,y)) ≤max{θ^A×B(m,n),θ^A×B(x,y)} =max{η^A×B(m,n)eiφ^A×B(m,n),η^A×B(x,y)eiφ^A×B(x,y)} =max{η^A×B(m,n),η^A×B(x,y)}eimax{φ^A×B(m,n),φ^A×B(x,y)}(AsA×Bis homogeneous)η^A×B((m,n)−(x,y)) ≤max{η^A×B(m,n),η^A×B(x,y)}φ^A×B((m,n)(x,y)) ≤max{φ^A×B(m,n),φ^A×B(x,y)}
Consequently, A×B¯ is IFSR and A×B̲ is π-IFSR.
Conversely, suppose that A×B¯ and A×B̲ is IFSR and π-IFSR, respectively. Then we have
ηA×B((m,n)−(x,y))≥min{ηA×B(m,n),ηA×B(x,y)}ηA×B((m,n)(x,y))≥min{ηA×B(m,n),ηA×B(x,y)}η^A×B((m,n)−(x,y))≤max{η^A×B(m,n),η^A×B(x,y)}η^A×B((m,n)(x,y))≤max{η^A×B(m,n),η^A×B(x,y)}φA×B((m,n)−(x,y))≥min{φA×B(m,n),φA×B(x,y)}φA×B((m,n)(x,y))≥min{φA×B(m,n),φA×B(x,y)}φ^A×B((m,n)−(x,y))≤max{φ^A×B(m,n),φ^A×B(x,y)}φ^A×B((m,n)(x,y))≤max{φ^A×B(m,n),φ^A×B(x,y)}ConsiderθA×B((m,n)−(x,y))=ηA×B((m,n)−(x,y))eiφA×B((m,n)−(x,y))≥min{ηA×B(m,n),ηA×B(x,y)}eimin{φA×B(m,n),φA×B(x,y)}=min{ηA×B(m,n)eiφA×B(m,n),ηA×B(x,y)eiφA×B(x,y)}=min{θA×B(m,n),θA×B(x,y)}Also,θA×B((m,n)(x,y))=ηA×B((m,n)(x,y))eiφA×B((m,n)(x,y))≥min{ηA×B(m,n),ηA×B(x,y)}eimin{φA×B(m,n),φA×B(x,y)}=min{ηA×B(m,n)eiφA×B(m,n),ηA×B(x,y)eiφA×B(x,y)}=min{θA×B(m,n),θA×B(x,y)}Moreover, θ^A×B((m,n)−(x,y))=η^A×B((m,n)−(x,y))eiφ^A×B((m,n)−(x,y))≤max{η^A×B(m,n),η^A×B(x,y)}eimax{φ^A×B(m,n),φ^A×B(x,y)}=max{η^A×B(m,n)eiφ^A×B(m,n),η^A×B(x,y)eiφ^A×B(x,y)}=max{θ^A×B(m,n),θ^A×B(x,y)}Also,θ^A×B((m,n)(x,y))=η^A×B((m,n)(x,y))eiφ^A×B((m,n)(x,y))≤max{η^A×B(m,n),η^A×B(x,y)}eimax{φ^A×B(m,n),φ^A×B(x,y)}=max{η^A×B(m,n)eiφ^A×B(m,n),η^A×B(x,y)eiφ^A×B(x,y)}=max{θ^A×B(m,n),θ^A×B(x,y)}.
Hence, A×B is CIFSR. Thus concluded the proof.
4. PROPERTIES OF LEVEL SUBSETS OF DIRECT PRODUCT OF COMPLEX INTUITIONISTIC FUZZY SUBRING
This section is devoted to study the properties of level subrings of direct product of CIFSR. We initiate the concept level subset of the direct product of CIFSR and prove various impactful results of this phenomena.
Definition 4.1.
Let A×B be Cartesian product of CIFS A and B. Then for r∈0,1,andt∈[0,2π] the level subset of CIFS A×B of S1×S2 is defined by
A×Br,tr^,t^m,n=m,n∈S1×S2:ηAm,n≥r,φAm,n≥t,η^Am,n≤r^,φ^Am,n≤t^,
For t=t̂=0, we obtain the level subset A×Brr̂m,n=m,n∈P:ηAm,n≥r,η^Am,n≤r̂,
and for r=r̂=0, then we obtain the level subset A×Btt̂m,n=m,n∈S1×S2:,φAm,n≥t,φ^Am,n≤t̂.
Theorem 4.2.
Let A and B be two CIFS of rings S1 and S2. Then A×Br,tr̂,t̂=Ar,tr̂,t̂×Br,tr̂,t̂.
Proof.
Consider (m,n)∈A(r,t)(r̂,t̂)×B(r,t)(r̂,t̂)
⇔m∈A(r,t)(r^,t^)andn∈B(r,t)(r^,t^)⇔ηA(m)≥r,φA(m)≥t,η^A(m)≤r^,φ^A(m)≤t^,andηB(n)≥r,φB(n)≥t,η^A(n)≤r^,φ^A(n)≤t^⇔min{ηA(m),ηB(n)}≥r,max{η^A(m),η^B(n)}≤r^min{φA(m),φB(n)}≥t,max{η^A(m),η^B(n)}≤t^⇔ηA×B(m,n)≥r,φA×B(m,n)≥t,andη^A×B(m,n)≤r^φ^A×B(m,n)≤t^⇔(m,n)∈(A×B)(r,t)(r^,t^)
Hence, (A×B)(r,t)(r̂,t̂)=A(r,t)(r̂,t̂)×B(r,t)(r̂,t̂).
Theorem 4.3.
Let A×B be CIFSR of ring S1×S2. Then (A×B)(r,t)(r̂,t̂) is a subring of ring S1×S2, for all r,r̂∈[0,1], and t,t̂∈[0,2π], where ηA(0,0′)≥r,φA(0,0′)≥t, η^A(0,0′)≤r̂, and φ^A(0,0′)≤t̂ also (0,0′) is identity element of S1×S2.
Proof.
Note that (A×B)(r,t)(r̂,t̂) is nonempty, as (0,0′)∈(A×B)(r,t)(r̂,t̂). Let (m,n),(x,y)∈(A×B)(r,t)(r̂,t̂) be any two elements. Then
ηA×B(m,n)≥r,φA×B(m,n)≥t,η^A×B(m,n)≤r^φ^A×B(m,n)≤t^,ηA×B(x,y)≥r,φA×B(x,y)≥t,η^A×B(x,y)ConsiderηA×B((m,n)−(x,y))≤r^φ^A×B(x,y)≤t^.eiφA×B((m,n)−(x,y))=θA×B((m,n)−(x,y))≥min{θA×B(m,n),θA×B(x,y)}
=min{ηA×B(m,n)eiφA×B(m,n),ηA×B(x,y)eiφA×B(x,y)}=min{ηA×B(m,n),ηA×B(x,y)}eimin{φA×B(m,n),φA×B(x,y)}(AsA×Bis homogeneous)ηA×B((m,n)−(x,y))≥min{ηA×B(m,n),ηA×B(x,y)}=min{r,r}=r.φA×B((m,n)−(x,y))≥min{φA×B(m,n),φA×B(x,y)}=min{t,t}=t.Moreover,η^A×B((m,n)−(x,y))eiφ^A×B((m,n)−(x,y))=θ^A×B((m,n)−(x,y))≤max{θ^A×B(m,n),θ^A×B(x,y)}=max{η^A×B(m,n)eiφ^A×B(m,n),η^A×B(x,y)eiφ^A×B(x,y)}=max{η^A×B(m,n),η^A×B(x,y)}eimax{φ^A×B(m,n),φ^A×B(x,y)}.(By homogeneous property)η^A×B((m,n)−(x,y))≤max{η^A×B(m,n),η^A×B(x,y)}=max{r,r}=r,φ^A×B((m,n)−(x,y))≤max{φ^A×B(m,n),φ^A×B(x,y)}=max{t,t}=t.Implies that(m,n)−(x,y)∈(A×B)(r,t)(r^,t^).Also,ηA×B((m,n)(x,y))eiφA×B((m,n)(x,y))=θA×B((m,n)(x,y))≥min{θA×B(m,n),θA×B(x,y)}=min{ηA×B(m,n)eiφA×B(m,n),ηA×B(x,y)eiφA×B(x,y)}=min{ηA×B(m,n),ηA×B(x,y)}eimin{φA×B(m,n),φA×B(x,y)}(By homogeneous property)ηA×B((m,n)(x,y))≥min{ηA×B(m,n),ηA×B(x,y)}=min{r,r}=r,and,φA×B((m,n)(x,y))≥min{φA×B(m,n),φA×B(x,y)}=min{t,t}=t,η^A×B((m,n)(x,y))eiφ^A×B((m,n)(x,y))=θ^A×B((m,n)(x,y))≤max{θ^A×B(m,n),θ^A×B(x,y)}=max{η^A×B(m,n)eiφ^A×B(m,n),η^A×B(x,y)eiφ^A×B(x,y)}=max{η^A×B(m,n),η^A×B(x,y)}eimax{φ^A×B(m,n),φ^A×B(x,y)}.(AsA×Bis homogeneous)η^A×B((m,n)(x,y))≤max{η^A×B(m,n),η^A×B(x,y)}=max{r,r}=r,and,φ^A×B((m,n)(x,y))≤max{φ^A×B(m,n),φ^A×B(x,y)}=max{t,t}=t.
Implies that (m,n)(x,y)∈(A×B)(r,t)(r̂,t̂). Hence (A×B)(r,t)(r̂,t̂) is subring.
Theorem 4.4.
Let A×B(r,t)(r̂,t̂) be a subring of ring S1×S2, then A×B is CIFSR of S1×S2, for all r∈[0,1], and t∈[0,2π], where ηA(0,0′)≥r,φA(0,0′)≥t, η^A(0,0′)≤r̂,φ^A(0,0′)≤t̂, also (0,0′) is identity element of S1×S2.
Proof.
Assume that min{ηA×B(m,n),ηA×B(x,y)}=r, min{φA×B(m,n),φA×B(x,y)}=t, max{η^A×B(m,n),η^A×B(x,y)}=r̂ and max{φ^A×B(m,n),φ^A×B(x,y)}=t̂. Then we have
ηA×B(m,n)≥r,φA×B(m,n)≥t,η^A×B(m,n)≤r^,φ^A×B(m,n)≤t^,ηA×B(x,y)≥r,φA×B(x,y)≥t,η^A×B(x,y)≤r^andφ^A×B(x,y)≤t^.This implies that(m,n)∈(A×B)(r,t)(r^,t^)and(x,y)∈(A×B)(r,t)(r^,t^).
As (A×B)(r,t)(r̂,t̂) is subring. So (m,n)−(x,y),(m,n)(x,y)∈(A×B)(r,t)(r̂,t̂). Then we have
ηA×B((m,n)−(x,y))≥r,φA×B((m,n)−(x,y))≥t,ηA×B((m,n)(x,y))≥r,φA×B((m,n)(x,y))≥t,η^A×B((m,n)−(x,y))≤r̂,φ^A×B((m,n)−(x,y))≤t̂,η^A×B((m,n)(x,y))≤r̂,φ^A×B((m,n)(x,y))≤t̂.Implies thatηA×B((m,n)−(x,y))≥min{ηA×B(m,n),ηA×B(x,y)}φA×B((m,n)−(x,y))≥min{φA×B(m,n),φA×B(x,y)},ηA×B((m,n)(x,y))≥min{ηA×B(m,n),ηA×B(x,y)},φA×B((m,n)(x,y))≥min{φA×B(m,n),φA×B(x,y)},η^A×B((m,n)−(x,y))≤max{η^A×B(m,n),η^A×B(x,y)},φ^A×B((m,n)−(x,y)≤max{φ^A×B(m,n),φ^A×B(x,y)},η^A×B((m,n)(x,y))≤max{η^A×B(m,n),η^A×B(x,y)},φ^A×B((m,n)(x,y))≤max{φ^A×B(m,n),φ^A×B(x,y)},ConsiderθA×B((m,n)−(x,y))=ηA×B((m,n)(x,y))eiηA×B((m,n)(x,y))≥min{ηA×B(m,n),ηA×B(x,y)}eimin{φA×B(m,n),φA×B(x,y)}=min{ηA×B(m,n)eiφA×B(m,n),ηA×B(x,y)eiφA×B(x,y)}θA×B((m,n)−(x,y))≥min{θA×B(m,n),θA×B(x,y)}Further,θA×B((m,n)(x,y))=ηA×B((m,n)(x,y))eiηA×B((m,n)(x,y))≥min{ηA×B(m,n),ηA×B(x,y)}eimin{φA×B(m,n),φA×B(x,y)}=min{ηA×B(m,n)eiφA×B(m,n),ηA×B(x,y)eiφA×B(x,y)}θA×B((m,n)(x,y))≥min{θA×B(m,n),θA×B(x,y)}Moreover,θ^A×B((m,n)−(x,y))=η^A×B((m,n)(x,y))eiη^A×B((m,n)(x,y))≤max{η^A×B(m,n),η^A×B(x,y)}eimax{φ^A×B(m,n),φ^A×B(x,y)}=max{η^A×B(m,n)eiφ^A×B(m,n),η^A×B(x,y)eiφ^A×B(x,y)}θ^A×B((m,n)−(x,y))≤max{θ^A×B(m,n),θ^A×B(x,y)}Also,θ^A×B((m,n)(x,y))=η^A×B((m,n)(x,y))eiη^A×B((m,n)(x,y))≤max{η^A×B(m,n),η^A×B(x,y)}eimax{φ^A×B(m,n),φ^A×B(x,y)}=max{η^A×B(m,n)eiφ^A×B(m,n),η^A×B(x,y)eiφ^A×B(x,y)}θ^A×B((m,n)(x,y))≤max{θ^A×B(m,n),θ^A×B(x,y)}.
Corollary 4.5.
Let A×B be a CIFSR of S1×S2, then the level subsets (A×B)rr̂ and (A×B)tt̂ is a subring of ring S1×S2, for all r,r̂∈[0,1], and t,t̂∈[0,2π], where ηA(0,0′)≥r,φA(0,0′)≥t, η^A×B(m,n)≤r̂, and φ^A×B(m,n)≤t̂ and (0,0′) is identity element of S1×S2.
5. HOMOMORPHISM OF DIRECT PRODUCT COMPLEX INTUITIONISTIC FUZZY SUBRING
In this section, we define the homomorphic image and preimage of the direct product of CIFSR. We prove some results of the direct product of CIFSR under ring homomorphism.
Definition 5.1.
Let f:S1×S2→R1×R2 be a ring homomorphism form S1×S2 to R1×R2. Let A×B and C×D be two CIFSR of rings S1×S2 and R1×R2, respectively. For all (x,y)∈S1×S2 and for all (m,n)∈R1×R2. The set
fA×B=m,n,fθA×Bm,n,fθ^A×Bm,n is said to be image of A×B, where
fθA×Bm,n=supθA×Bx,y,iffx,y=m,n,f−1(m,n)≠∅0,otherwise
fθ^A×Bm,n=infθ^A×Bx,y,iffx,y=m,n,f−1(m,n)≠∅1,
The set f−1B1×B2=x,y,f−1θC×Dx,y,f−1θ^C×Dx,y is said to be preimage of C×D, where
f−1θC×Dx,y=θC×Dfx,y,
f−1θ^C×Dx,y=θC×Dfx,y.
Theorem 5.2.
[9] f:S→R be ring homomorphism from S to R. Let A be IFSR of S and B be IFSR of R. Then f(A) is IFSR of R and f−1(B) is IFSR of S.
Lemma 5.3.
Let f:S1×S2→R1×R2 be a homomorphism from group S1×S2 to group R1×R2. Let A×B and C×D be two CIFSR. Then
fθA×Bm,n=fηA×Bm,neifφA×Bm,n,forall(m,n)∈R1×R2
fθ^A×Bm,n=fη^A×Bm,neifφ^A×Bm,n
f−1θC×Dx,y=f−1ηC×Dx,yeif−1φC×Dx,y,forall(x,y)∈S1×S2
f−1θ^C×D(x,y)=f−1η^C×Dx,yeif−1φ^C×Dx,y
Proof.
Considerf(θA×B)(m,n)=max{(θA×B)(x,y),iff(x,y)=(m,n)}=max{(ηA×B)(x,y)eif(φA×B)(x,y),iff(x,y)=(m,n)}=max{(ηA×B)(x,y),iff(x,y)=(m,n)}eimax{(φA×B)(x,y),iff(x,y)=(m,n)}=f(ηA×B)(m,n)eif(φA×B)(m,n).
Hence, f(θA×B)(m,n)=f(ηA×B)(m,n)eif(φA×B)(m,n).
Suppose thatf(θ^A×B)(m,n)=max{(θ^A×B)(x,y),iff(x,y)=(m,n)}=max{(η^A×B)(x,y)eif(φ^A×B)(x,y),iff(x,y)=(m,n)}=max{(η^A×B)(x,y),iff(x,y)=(m,n)}eimax{(φ^A×B)(x,y),iff(x,y)=(m,n)}=f(η^A×B)(m,n)eif(φ^A×B)(m,n).
As a result, f(θ^A×B)(m,n)=f(η^A×B)(m,n)eif(φ^A×B)(m,n).
Assume that,f−1(θC×D)(x,y)=θC×D(f(x,y))=ηC×D(f(x,y))eiφC×D(f(x,y))=f−1(ηC×D)(x,y)×eif−1(φC×D)(x,y).
Consequently, f−1(θC×D)(x,y)=f−1(ηC×D)(x,y)eif−1(φC×D)(x,y).
Consider,f−1(θ^C×D)(x,y)=θ^C×D(f(x,y))=η^C×D(f(x,y))eiφ^C×D(f(x,y))=f−1(η^C×D)(x,y)×eif−1(φ^C×D)(x,y).
Consequently, f−1(θ^C×D)(x,y)=f−1(η^C×D)(x,y)eif−1(φ^C×D)(x,y).
Theorem 5.4.
Let f:S1×S2→R1×R2 be a homomorphism from ring S1×S2 to ring R1×R2. Let A×B be CIFSR of S1×S2. Then f(A×B) is CIFSR of R1×R2.
Proof.
Obviously, {((p,q),ηA×B(p,q),η^A×B(p,q)),(p,q)∈S1×S2} and {((p,q),φA×B(p,q),φ^A×B(p,q),(p,q)∈S1×S2)} are IFSR and π-IFSR, respectively. From Theorem 2.9 and Theorem 5.2 the homomorphic image of {((p,q),ηA×B(p,q),η^A×B(p,q)),(p,q)∈S1×S2} and {((p,q),φA×B(p,q),φ^A×B(p,q),(p,q)∈S1×S2)} are IFSR and π-IFSR, respectively, for all (m,n),(x,y)∈R1×R2. Then we have
f(ηA×B)((m,n)−(x,y))≥min{f(ηA×B)(m,n),f(ηA×B)(x,y)},f(ηA×B)((m,n)(x,y))≥min{f(ηA×B)(m,n),f(ηA×B)(x,y)},f(η^A×B)((m,n)−(x,y))≤max{f(η^A×B)(m,n),f(η^A×B)(x,y)},f(η^A×B)((m,n)(x,y))≤max{f(η^A×B)(m,n),f(η^A×B)(x,y)},f(φA×B)((m,n)−(x,y))≥min{f(φA×B)(m,n),f(φA×B)(x,y)},f(φA×B)((m,n)(x,y))≥min{f(φA×B)(m,n),f(φA×B)(x,y)},f(φ^A×B)((m,n)−(x,y))≤max{f(φ^A×B)(m,n),f(φ^A×B)(x,y)},f(φ^A×B)((m,n)(x,y))≤max{f(φ^A×B)(m,n),f(φ^A×B)(x,y)},From Lemma 5.3,we havef(θA×B)((m,n)−(x,y))=f(ηA×B)((m,n)−(x,y))eif(φA×B)((m,n)−(x,y)),forall(m,n),(x,y)∈R1×R2≥min{f(ηA×B)(m,n),f(ηA×B)(x,y)}ei{f(φA×B)(m,n),f(φA×B)(x,y)}=min{f(ηA×B)(m,n)eif(φA×B)(m,n),f(ηA×B)(y2)eif(φA×B)(x,y)}=min{f(θA×B)(m,n),f(θA×B)(x,y)}Consequently,f(θA×B)((m,n)−(x,y))≥min{f(θA×B)(m,n),f(θA×B)(x,y)}Further,f(θA×B)((m,n)(x,y))=f(ηA×B)((m,n)(x,y))eif(φA×B)((m,n)(x,y)),forall(m,n),(x,y)∈R1×R2≥min{f(ηA×B)(m,n),f(ηA×B)(x,y)}ei{f(φA×B)(m,n),f(φA×B)(x,y)}=min{f(ηA×B)(m,n)eif(φA×B)(m,n),f(ηA×B)(y2)eif(φA×B)(x,y)}=min{f(θA×B)(m,n),f(θA×B)(x,y)}.As result,f(θA×B)((m,n)(x,y))≥min{f(θA×B)(m,n),f(θA×B)(x,y)}.In the view of Lemma 5.3,we know thatf(θ^A×B)((m,n)−(x,y))=f(η^A×B)((m,n)−(x,y))eif(φ^A×B)((m,n)−(x,y)),(m,n),(x,y)∈R1×R2≤max{f(η^A×B)(m,n),f(η^A×B)(x,y)}ei{f(φ^A×B)(m,n),f(φ^A×B)(x,y)}=max{f(η^A×B)(m,n)eif(φ^A×B)(m,n),f(η^A×B)(y2)eif(φ^A×B)(x,y)}=max{f(θ^A×B)(m,n),f(θ^A×B)(x,y)}Consequently,f(θ^A×B)((m,n)−(x,y))≤max{f(θ^A×B)(m,n),f(θ^A×B)(x,y)}.Moreover,f(θ^A×B)((m,n)(x,y))=f(η^A×B)((m,n)(x,y))eif(φ^A×B)((m,n)(x,y)),(m,n),(x,y)∈R1×R2≤max{f(η^A×B)(m,n),f(η^A×B)(x,y)}ei{f(φ^A×B)(m,n),f(φ^A×B)(x,y)}=max{f(η^A×B)(m,n)eif(φ^A×B)(m,n),f(η^A×B)(y2)eif(φ^A×B)(x,y)}=max{f(θ^A×B)(m,n),f(θ^A×B)(x,y)}Therefore,f(θ^A×B)((m,n)−(x,y))≤max{f(θ^A×B)(m,n),f(θ^A×B)(x,y)}.
This establishes the proof.
Theorem 5.5.
Let f:S1×S2→R1×R2 be a homomorphism from ring S1×S2 to ring R1×R2. Let C×D be CIFSR of R1×R2. Then f−1(C×D) is CIFSR of S1×S2.
Proof.
Obviously, {((p,q),ηC×D(p,q),η^C×D(p,q)),(p,q)∈R1×R2} and {((p,q),φC×D(p,q),φ^C×D(p,q),(p,q)∈R1×R2)} are IFSR and π-IFSR respectively. Then from Theorems 2.9 and 5.2 the inverse image of {((p,q),ηC×D(p,q),η^C×D(p,q)),(p,q)∈R1×R2} and {((p,q),φC×D(p,q),φ^C×D(p,q),(p,q)∈R1×R2)} are IFSR and π-IFSR, respectively, for all (m,n),(x,y)∈S1×S2. Then we have
f−1(ηC×D)((m,n)−(x,y)) ≥min{f−1(ηC×D)(m,n),f−1(ηC×D)(x,y)},f−1(ηC×D)((m,n)(x,y)) ≥min{f−1(ηC×D)(m,n),f−1(ηC×D)(x,y)},f−1(η^C×D)((m,n)−(x,y)) ≤max{f−1(η^C×D)(m,n),f−1(η^C×D)(x,y)},f−1(η^C×D)((m,n)(x,y)) ≤max{f−1(η^C×D)(m,n),f−1(η^C×D)(x,y)},f−1(φC×D)((m,n)−(x,y)) ≥min{f−1(φC×D)(m,n),f−1(φC×D)(x,y)},f−1(φC×D)((m,n)(x,y)) ≥min{f−1(φC×D)(m,n),f−1(φC×D)(x,y)},f−1(φ^C×D)((m,n)−(x,y)) ≤max{f−1(φ^C×D)(m,n),f−1(φ^C×D)(x,y)},f−1(φ^C×D)((m,n)(x,y)) ≤max{f−1(φ^C×D)(m,n),f−1(φ^C×D)(x,y)}.
By Lemma 5.3(3), we havef−1(θC×D)((m,n)−(x,y)) =f−1(ηC×D)((m,n)−(x,y))eif−1(φC×D)((m,n)−(x,y)), forall(m,n),(x,y)∈S1×S2 ≥min{f−1(ηC×D)(m,n),f−1(ηC×D)(x,y) eif−1(φC×D)(m,n),f−1(φC×D)(x,y) =min{f−1(ηC×D)(m,n)eif−1(φC×D)(m,n), f−1(ηC×D)(x,y)eif−1(φC×D)(x,y) min{f−1(θC×D)(m,n),f−1(θC×D)(x,y)}.Therefore,f−1(θC×D)((m,n)−(x,y)) ≥min{f−1(θC×D)(m,n),f−1(θC×D)(x,y)}.
Further, f−1(θC×D)((m,n)(x,y))=f−1(ηC×D)((m,n)(x,y))eif−1(φC×D)((m,n)(x,y)),forall(m,n),(x,y)∈S1×S2.
≥min{f−1(ηC×D)(m,n),f−1(ηC×D)(x,y)}ei{f−1(φC×D)(m,n),f−1(φC×D)(x,y)}=min{f−1(ηC×D)(m,n)eif−1(φC×D)(m,n),f−1(ηC×D)(x,y)eif−1(φC×D)(x,y)}=min{f−1(θC×D)(m,n),f−1(θC×D)(x,y)}.Therefore,f−1(θC×D)((m,n)(x,y))≥min{f−1(θC×D)(m,n),f−1(θC×D)(x,y)}
Consider f−1(θ^C×D)((m,n)−(x,y))=f−1(η^C×D)((m,n)−(x,y))eif−1(φ^C×D)((m,n)−(x,y)),forall(m,n),(x,y)∈S1×S2
≤max{f−1(η^C×D)(m,n),f−1(η^C×D)(x,y)}ei{f−1(φ^C×D)(m,n),f−1(φ^C×D)(x,y)}=max{f−1(η^C×D)(m,n)eif−1(φ^C×D)(m,n),f−1(η^C×D)(x,y)eif−1(φ^C×D)(x,y)}=max{f−1(θ^C×D)(m,n),f−1(θ^C×D)(x,y)}
Therefore, f−1(θ^C×D)((m,n)−(x,y))≤maxf−1(θ^C×D)(m,n),f−1(θ^C×D)(x,y). Further, f−1(θ^C×D)((m,n)(x,y))
=f−1(η^C×D)((m,n)(x,y))eif−1(φ^C×D)((m,n)(x,y)),forall(m,n),(x,y)∈S1×S2≤max{f−1(η^C×D)(m,n),f−1(η^C×D)(x,y)}ei{f−1(φ^C×D)(m,n),f−1(φ^C×D)(x,y)}=max{f−1(η^C×D)(m,n)eif−1(φ^C×D)(m,n),f−1(η^C×D)(x,y)eif−1(φ^C×D)(x,y)}=max{f−1(θ^C×D)(m,n),f−1(θ^C×D)(x,y)}
Consequently, f−1(θ^C×D)((m,n)(x,y))≤max{f−1(θ^C×D)(m,n),f−1(θ^C×D)(x,y)}. This concluded the proof.
, M. Haris Mateen2,