T-Fuzzy Structure on JU-Algebra

Definition 1.

⁴ A JU-algebra is an algebra of (X, ◦, 1) of type (2, 0) with a binary operation ◦ and a fixed element 1, if it holds, ∀ x, y, z ∈ X:

In what follows, let (X, ◦, 1) denote a JU-algebra unless otherwise specified. For brevity, we also refer to call X JU-algebra.

Lemma 1.

^4,15 If X is a JU-algebra, then x ◦ x = 1 for any x ∈ X .

Definition 2.

⁵ A nonempty subset S of a JU-algebra X is called JU-subalgebra of X, if x ◦ y ∈ S, ∀x, y ∈ S .

Definition 3.

⁵A nonempty subset J of a JU-algebra X is said to be a JU-ideals of X if it satisfies:

Definition 4.

^16,17 A fuzzy set ϖ in a set X is a pair (X, M _ϖ), where the function M _ϖ: X → [0, 1] is called the memebership function of ϖ . For γ ∈ [0, 1], the set U(M _ϖ; γ) = {x ∈ X|M _ϖ ≥ γ} is called an upper level subset of ϖ .

Definition 5.

¹⁴ A triangular norm(T-norm) is a function T: [0, 1] × [0, 1] → [0, 1] that satisfies the following conditions:

Examples of T-norm are:

Lemma 2.

¹⁸ Let $T$ and $T^{\prime }$ be T-norms. Then T′ (T (p, q), T (r, s)) = T (T′ (p, r), T′ (q, s))

Definition 6.

¹⁹ In a JU-algebra (X, ◦, 1), an element x ∈ X is idempotent if x ◦ x = x .

Definition 7.

¹⁴ Let T be a T-norm, denoted by T _{i d e m} the set of all idempotent with respect to T, That is, T _{i d e m} = {T (x, x) = x , for some x ∈ [0, 1]}. A fuzzy set ϖ in X is said to be an idempotent T-fuzzy set if Im(M _ϖ) ⊆ T _{i d e m}.

Definition 8.

Let ϖ = (X, M _ϖ) be a fuzzy set in X. Then the set ϖ is a T-fuzzy JU-subalgebra with the binary operation ◦ if

Example 1.

Let X = {1, 2, 3, 4} in which ◦ is defined by the following Cayley table: “See ( Table 1)⁴ is a JU-algebra”. Let T: [0, 1] × [0, 1] → [0, 1] be a function. The given T-norm defined by T (x, y) = max{x + y − 1, 0}, ∀x, y ∈ [0, 1]. Define a fuzzy set ϖ in X by M _ϖ (1) = 0.8, M _ϖ (2) = 0.6, M _ϖ (3) = 0.5 and M _ϖ (4) = 0.3. By routine calculation ϖ is a T-fuzzy JU-subalgebra of X.

Theorem 3.

If ϖ is an idempotent T-fuzzy JU-subalgebra of X, then M _ϖ (1) ≥ M _ϖ (x).

Proof.

Suppose ϖ is an idempotent T-fuzzy JU-subalgebra. Since M _ϖ (1) = M _ϖ (x ◦ x) ≥ T{M _ϖ (x), M _ϖ (x)} = M _ϖ (x).

Theorem 4.

The intersection of any two T-fuzzy JU-subalgebras of X is also a T-fuzzy JU-subalgebra of X.

Proof.

Suppose ϖ₁ = (X, M _ϖ1) and ϖ₂ = (X, M _ϖ2) are two T-fuzzy JU-subalgebras of X and ∀x, y ∈ X. Then,

Corollary 5.

Let { _j: j ∈ Ω} be a family of T-fuzzy JU-subalgebras of X. Then ∩ _{j ∈Ω}ϖ _j is also T-fuzzy JU-subalgebra of X, where ∩ _{j ∈Ω}ϖ _j = {(x, i n f _{j ∈Ω} M _ϖj) |x ∈ X}.

Remark 1.

The union of any two T-fuzzy JU-subalgebras of a JU-algebra X may not be a T-fuzzy JU-subalgebra of X.

Example 2.

Let X = {1, 2, 3, 4} in which ◦ is defined by the following Cayley table: “(See Table 2)⁴ is a JU-algebra”. Let T: [0, 1] × [0, 1] → [0, 1] be a function. The given T-norm defined by T (x, y) = min{x, y}, ∀x, y ∈ [0, 1]. Let us define the fuzzy set ϖ₁ and ϖ₂ in X by

M _ϖ1 (1) = 0.7, M _ϖ1 (2) = 0.4, M _ϖ1 (3) = 0.2, M _ϖ1 (4) = 0.1 and

M _ϖ2 (1) = 0.8, M _ϖ2 (2) = 0.6, M _ϖ2 (3) = 0.3 and M _ϖ2 (4) = 0.2.

Then, M _{ϖ1 ∪ ϖ2} (2 ◦ 3) = max{M _ϖ1 (2 ◦ 3), M _ϖ2 (2 ◦ 3)} = 0.2                (*)

And, M _{ϖ1 ∪ ϖ2} (2 ◦ 3) = max{M _ϖ1 (2 ◦ 3), M _ϖ2 (2 ◦ 3)}

         ≥ max{T (M _ϖ1 (2), M _ϖ1 (3)), T (M _ϖ2 (2), M _ϖ2 (3))}

         = T{max{M _ϖ1 (2), M _ϖ2 (2)}, max{M _ϖ1 (3), M _ϖ2 (3)}} = 0.3         (**)

From (∗) and (∗∗) we get 0.2 ≥ 0.3 which is false.

Theorem 6.

Let S be a nonempty subest of a JU-algebra X. Then the characteristics function X _S is a T-fuzzy JU-subalgebras of X if and only if S is a subalgebra of X.

Proof.

Suppose X _S is a T-fuzzy JU-subalgebras of X and S ≠ ∅. Let x, y ∈ S implies that

X _S (x) = 1 = X _S (y). Now

X _J (x ◦ y) ≥ T{X _S (x), X _S (y)} = T{1, 1} = 1

→ X _S (x ◦ y) ≥ 1 but X _S (x ◦ y) ≤ 1

→ X _S (x ◦ y) = 1

→ x ◦ y ∈ S

→ S is the subalgebra of X.

Conversely, suppose S is a JU-subalgebra of X. We need to show that X _S is a T-fuzzy JU-subalgebra of X. Now consider the following cases:

Case 1: if x, y ∈ X where x ◦ y ∈ S then

X _S (x ◦ y) = 1 ≥ T{X _S (x), X _S (y)}

Case 2: if x ∈ S and y ∉ (or x ∉ S and y ∈ S), then

X _S (x) = 1, X _S (y) = 0. Thus

X _S (x ◦ y) ≥ 0 = T{1, 0} = T{0, 1} = T{X _S (x), X _S (y)}

Case 3: Suppose x, y ∉ S then

X _S (x) = 0 = X _S (y). Thus

X _S (x ◦ y) ≥ 0 = T{0, 0} = T{X _S (x), X _S (y)}

Theorem 7.

Let U(M _ϖ: γ) be nonempty and ∀γ ∈ [0, 1]. A fuzzy subset ϖ of a JU-algebra X is T-fuzzy JU-subalgebra of X, if and only if U(M _ϖ: γ) is subalgebra of X.

Proof.

Suppose that ϖ is T-fuzzy JU-subalgebra of X. Since X is JU-algebra, then 1 ∈ X implies that M _ϖ (1) ≥ γ.

$\to 1\in U(M:\gamma )$

$\to U(M_{\varpi }:\gamma )\ne \varnothing ,\gamma \in [0,1].$

Let x, y ∈ U(M: γ). Then, M _ϖ (x) ≥ γ and M _ϖ (y) ≥ γ. We have

This implies that x ◦ y ∈ U(M _ϖ: γ) and hence U(M _ϖ: γ) is a subalgebra of X. Conversely, suppose that U(M _ϖ: γ) is a JU-subalgebra of X for any γ ∈ [0, 1] and U(M _ϖ: γ) ≠ ∅. Assume that ϖ is not T-fuzzy JU-subalgebra of X. Then there exist some z₀, z₁ ∈ X such that

Take β = $\frac{1}{2}$ [M _ϖ (z₀ ◦ z₁) + {M _ϖ (z₀), M _ϖ (z₁)}]

$\to M_{\varpi }(z_0◦z_1)<\beta <\{M_{\varpi }\left(z_0\right),M_{\varpi }\left(z_1\right)\}$

→ z₀ ◦ z₁ ∉ (M: β), a contradiction, since U(M _ϖ: β) is a subalgebra of X. Therefore, M _ϖ (z₀ ◦ z₁) ≥ {M _ϖ (z₀), M _ϖ (z₁)} for any z₀, z₁ ∈ X.

Theorem 8.

Let T be a T-norm and let ϖ = (X, M _ϖ) be a fuzzy set in a JU-algebra of X with I(M _ϖ) = {γ₁, γ₂, …, γ _n} where γ _i < γ _j whenever i > j . Assume that there exist an ascending chain of subalgebra S _o ⊆ S₁ ⊆ S₂ ⊆, …, ⊆ S _n = X of X such that M _ϖ ( ${\tilde{S}}_m$ ) = γ _m, where ${\tilde{S}}_m$ = S _m/S _m−1 for m = 1, 2, 3, n and ${\tilde{S}}_o$ = S _o. Then ϖ is a T-fuzzy JU-subalgebra of X.

Proof.

Suppose that there exist an ascending chain of subalgebra S _o ⊆ S₁ ⊆ S₂ ⊆, …, ⊆ S _n = X of X such that M _ϖ ( ${\tilde{S}}_m$ ) = γ _m, where m = 1, 2, 3,…, n. Let x, y ∈ X and let x, y ∈ ${\tilde{S}}_m$ then M _ϖ (x) = γ _m = M _ϖ (y) and x ◦ y ∈ ${\tilde{S}}_m$ . Now

Suppose that x ∈ ${\tilde{S}}_p$ and y ∈ ${\tilde{S}}_q$ for p ≠ q without loss of of generality we may assume that p > q. Then M _ϖ (x) = γ _p < γ _q = M _ϖ (y) and x ◦ y ∈ $\tilde{S}$ . Thus

Hence ϖ is a T-fuzzy JU-subalgebra of X.

Theorem 9.

let S be a JU-subalgebra of X and ϖ be a fuzzy set in X given by

With p ≥ q. Then ϖ is a T _{L u k} fuzzy JU-subalgebra of X. Particularly if p = 1 and q = 0 then ϖ is an idempotent T _{L u k} fuzzy subalgebra X.

Additionally, Im(ϖ) = S

Proof.

Let x, y ∈ X. Let us consider the following cases:

Case 1, If x, y ∈ S, then

Case 2, If x, y ∉ S, then

Case 3, If x ∈ S and y ∉ S (or x ∉ S and y ∈ S), then T _{L u k} ((x), M _ϖ (y)) = T _{L u k} (p, q)

Hence ϖ is T-fuzzy JU-subalgebra of X.

Suppose that p = 1 and q = 0. Then T _{L u k} (p, p) = m a x{p + p − 1, 0} = 1 = p and T _{L u k} (q, q) = {q + q − 1, 0} = 0 = q. Thus p, q ∈ (T _{i d e m}), and Im(M _ϖ) = S.

Definition 9.

Let ϖ be a fuzzy set in X. Then the set ϖ is a T-fuzzy JU-ideal with the binary operation ◦ if it satisfies the following axioms:

Example 3.

Let X = {1, 2, 3, 4, 5, 6} in which ◦ is defined by the following Cayley table” (See Table 3)⁴ is a JU-algebra”. Let T: [0, 1] × [0, 1] → [0, 1] be a function. The given T-norm defined by T (x, y) = x.y , for all x, y ∈ [0, 1]. Define a fuzzy set ϖ in X by M _ϖ (1) = 0.9, M _ϖ (2) = 0.7 = M _ϖ (3), M _ϖ (4) = 0.5 , M _ϖ (5) = 0.3 = M _ϖ (6). By routine calculation ϖ is a T-fuzzy JU-ideals of X. (i.e., M _ϖ (4) = 0.5 ≥ 0.45 = T{M _ϖ (1), M _ϖ (1 ◦ 4)} = T{0.9, 0.5} = (0.9).(0.5))

Theorem 10.

The intersection of any two T-fuzzy JU-ideals of X is also T-fuzzy JU-ideal of X.

Proof.

Suppose ϖ₁ = (X, M _ϖ1) and ϖ₂ = (X, M _ϖ2) are two T-fuzzy JU-ideals of X and ∀x, y ∈ X. Then,

And,

Corollary 11.

Let {ϖ _j: j ∈ Ω} be a family of T-fuzzy JU-ideals of X. Then ∩ _{j ∈Ω}ϖ _j is also T-fuzzy JU-ideal of X, where ∩ _{j ∈Ω}ϖ _j = {(x, i n f _{j ∈Ω} M _ϖj)|x ∈ X}.

Theorem 12.

Let J be a nonempty sub of a JU-algebra X. Then the characteristics function X _J is a T-fuzzy JU-ideals of X if and only if J is an ideal of X.

Proof.

Suppose X _J is a T-fuzzy JU-ideals of X and J ≠ ∅. Let x ∈ J implies that X _J (x) = 1.

Hence, X _J (1) = X _J (x ◦ x) ≥ T{X _J (x), X _J (x)} = T{1, 1} = 1

$\to X_J\left(1\right)\ge 1\mathrm{but}{X}_J\left(1\right)\le 1$

$\to X_J\left(1\right)=1$

$\to 1\in J$

And,

Let x, x ◦ y ∈ J implies that X _J (x) = 1 = X _J (x ◦ y). Now

$X_J\left(y\right)\ge T\{X_J\left(x\right),X_J(x◦y)\}=T\{1,1\}=1$

$\to X_J\left(y\right)\ge 1\mathrm{but}{X}_J\left(y\right)\le 1$

$\to X_J\left(y\right)=1$

$\to y\in J$

$\to X_J\left(y\right)=1$

→ J is a JU-ideal of X.

Conversely, suppose J is ideal of X. We need to show that X _J is a T-fuzzy JU-ideal of X. Now consider the following cases:

Case 1: if x, x ◦ y ∈ J where y ∈ J then

$X_J\left(y\right)=1\ge T\{X_J\left(x\right),X_J(x◦y)\}$

Case 2: if x ∈ J and x ◦ y ∉ (or x ∉ J and x ◦ y ∈ J), then

X _J (x) = 1, X _J (x ◦ y) = 0. Thus

$X_J\left(y\right)\ge 0=T\{1,0\}=T\{0,1\}=T\{X_J\left(x\right),X_J(x◦y)\}$

Case 3: if x, x ◦ y ∉ J then

X _J (x) = 0 = X _J (x ◦ y). Thus

$X_J\left(y\right)\ge 0=T\{0,0\}=T\{X_J\left(x\right),X_J(x◦y)\}$

Theorem 13.

Let (M _ϖ: γ) be nonempty and ∀γ ∈ [0, 1]. A fuzzy subset ϖ of a JU-algebra X is T-fuzzy JU-ideal of X, if and only if U(M _ϖ: γ) is JU-ideal of X.

Proof.

Suppose ϖ is a T-fuzzy JU-ideal of X. Let γ ∈ [0, 1] and (M _ϖ: γ) ≠ ∅.

Let x, x ◦ y ∈ (M _ϖ: γ) implies that M _ϖ (x) ≥ γ and M _ϖ (x ◦ y) ≥ γ. Then

This implies that y ∈ (M _ϖ: γ) and hence U(M _ϖ: γ) is a JU-ideal of X.

Conversely, suppose that (M _ϖ: γ) is a JU-ideal of X for any γ ∈ [0, 1] and U(M _ϖ: γ) ≠ ∅. Assume that ϖ is not T-fuzzy JU-ideal of X. Then there exist some z₀ ∈ X such that

z₀ ∈ U(M: β) and 1 ∉ U(M _ϖ: β), which is contradict to our assumption that U(M _ϖ: β) is a JU-ideal of X. Therefore, M _ϖ (1) ≥ M _ϖ (x), ∀x ∈ X. And assume that z₀, z₁ ∈ X such that

Take β = $\frac{1}{2}$ [M _ϖ (z₁) + {M _ϖ (z₀), M _ϖ (z₀ ◦ z₁)}]

z₁ ∉ U(M: β), a contradiction, since U(M _ϖ: β) is a JU-ideal of X.

Therefore, M _ϖ (z₁) ≥ {M _ϖ (z₀), M _ϖ (z₀ ◦ z₁)} for any z₀, z₁ ∈ X.

Theorem 14.

Let ϖ be an idempotent T-fuzzy JU-ideals of X. Then the set X _{M ϖ} = {x ∈ X|M _ϖ (x) = M _ϖ (1)} is an ideal of X.

Proof.

Let ϖ be an idempotent T-fuzzy JU-ideals of X. Obviously, 1 ∈ X _{M ϖ}. Let x, x ◦ y ∈ X _{M ϖ} implies that M _ϖ (x) = M _ϖ (1) = M _ϖ (x ◦ y). Now

→ M _ϖ (y) ≥ M _ϖ (1) but M _ϖ (y) ≤ M _ϖ (1) by definition 9(a)

$\to M_{\varpi }\left(y\right)=M_{\varpi }\left(1\right)$

$\to y\in {X_M}_{\varpi }$

Theorem 15.

If every T-fuzzy JU-ideal ϖ of X has a finite image, then every descending chain of JU-ideals of X converges at finite steps.

Proof.

Assume that there exists a strictly descending chain J₁ ⊋ J₂ ⊋ J₃ … of JU-ideal of X which does not converge at finite step. Define a fuzzy set ϖ in X by

Since 1 ∈ J _n, ∀ n, M _ϖ (1) = 1 ≥ M _ϖ (x), ∀x ∈ X.

And for any x, y ∈ X then by the above assumption consider the following cases.

Case 1. If x ∈ J _n \ J _n+1 and x ◦ y ∈ J _m \ J _m+1 for n = 1, 2, 3, …; m = 1, 2, 3, … Without loss of generality, we may assume that n ≤ m.

Then x and x ◦ y ∈ J _n, (since J _m ⊊ J _n). Thus y ∈ J _n since J _n is a JU-ideals of X. Hence,

Case 2. If x, x ◦ y ∈ ${\cap }_{n=1}^{\infty }$ J _n, then y ∈ ${\cap }_{n=1}^{\infty }$ J _n.

Thus M _ϖ (y) = 1 = T{M _ϖ (x), M _ϖ (x ◦ y)}.

Case 3. If x ∉ ${\cap }_{n=1}^{\infty }$ J _n and x ◦ y ∈ ${\cap }_{n=1}^{\infty }$ J _n, then there exists a positive integer p such that

It follows that y ∈, then M _ϖ (y) ≥ $\frac{p-1}{p}$ = T{M _ϖ (x), M _ϖ (x ◦ y)}.

Or, if x ∈ ${\cap }_{n=1}^{\infty }$ J _n and x ◦ y ∉ ${\cap }_{n=1}^{\infty }$ J _n, then there exists a positive integer q such that x ◦ y ∈ J _q \ J _q+1. Implies that y ∈, then M _ϖ (y) ≥ $\frac{q-1}{q}$ = T{M _ϖ (x), M _ϖ (x ◦ y)}.

Thus, ϖ is a T-fuzzy JU-ideals with an infinite number of different values, which is a contradiction.

Theorem 16.

Every ascending chain of JU-ideals of X terminates at finite step if and only if the set of values of any T-fuzzy JU-ideals is a well ordered subset of [0, 1].

Proof.

Let ϖ be a T-fuzzy JU-ideals of X. Assume that the set of values of ϖ is not a well-ordered subset of [0, 1]. Then there exist a strictly decreasing sequence {γ _n} such that M _ϖ (x _n) = γ _n. This implies that (M _ϖ: γ₁) ⊊ U(M _ϖ: γ₂) ⊊ U(M _ϖ: γ₃) ⊊ … is strictly ascending chain of the JU-ideal of X that does not terminate. This is impossible. In contrast, assume that a strictly ascending chain

We went to show that ϖ is a T-fuzzy JU-ideal of X. Since 1 ∈ J _n, ∀n, M _ϖ (1) ≥ $\frac{1}{n}$ = M _ϖ (x), ∀x ∈ X.

And for any x, y ∈ X then by the above assumption consider the following cases.

Case 1: if x, x ◦ y ∈ J _n \ J _n−1 for n = 2, 3, … then y ∈ J _n \ J _n−1 implies that $M_{\varpi }\left(y\right)\ge \frac{1}{n}=T\{M_{\varpi }\left(x\right),M_{\varpi }(x◦y)\}$

Case 2: If x ∈ J _n and x ◦ y ∈ J _n \ J _m and x ◦ y ∈ J _n for any m < n. Since ϖ is a JU-ideal of X, then y ∈ J _n.Thus M _ϖ (y) ≥ $\frac{1}{n}$ ≥ $\frac{1}{m+1}$ ≥ M _ϖ (x ◦ y).

Or, If x ∈ J _n \ J _m and x ◦ y ∈ J _n and x ∈ J _n for any m < n. Since ϖ is a JU-ideal of X, then y ∈ J _n.Thus M _ϖ (y) ≥ $\frac{1}{n}$ ≥ $\frac{1}{m+1}$ ≥ M _ϖ (x ◦ y).

Therefore, ϖ is a T-fuzzy JU-ideals of X. Since the sequence (*) is not converge, ϖ has a strictly descending sequence of values. This contradicts that the value set of any T-fuzzy JU-ideal is well-ordered.

Definition 10.

Let ϖ₁ = (X, M _ϖ1) and ϖ₂ = (Y, M _ϖ2) be two T-fuzzy JU-subalgebra of a JU-algebra X and Y respectively. The Cartesian product of ϖ₁ and ϖ₂ with respect to t-norm T denoted by $M_{{[{\varpi }_1\times {\varpi }_2]}_T}$ = (X × Y, $M_{{[{\varpi }_1\times {\varpi }_2]}_T}$ ), is defined by $M_{{[{\varpi }_1\times {\varpi }_2]}_T}$ (x, y) = T (M _ϖ1 (x), M _ϖ2 (y)) , ∀(x, y) ∈ X × Y .

Theorem 17.

The Cartesian product of any two T-fuzzy JU-subalgebras of X is also a T-fuzzy JU-subalgebra of X.

Proof.

Suppose ϖ₁ = (X, M _ϖ1) and ϖ₂ = (Y, M _ϖ2) be two T-fuzzy JU-subalgebras of a JU-algebra X. Let (x₁, x₂), (y₁, y₂) ∈ X × Y. Then

Theorem 18.

Let $\underset{j=1}{\overset{n}{\left\{X_j\right\}}}$ be the finite family of JU-algebras and X = $\underset{j=1}{\overset{n}{\pi }}$ X _j the Cartesian product of JU-algebras of {X _j}. Let ϖ _j be a T-fuzzy JU-subalgebra of X, where 1 ≤ j ≤ n . Then ϖ = $\underset{j=1}{\overset{n}{\pi }}$ ϖ _j is defined by M _ϖ (x₁, x₂, …, x _n) = $M_{{\left(\underset{j=1}{\overset{n}{\pi }}{\varpi }_j\right)}_T}$ (x₁, x₂, …, x _n) = T _n (M _{ϖ1 (x1)}, M _{ϖ2 (x2)}, …, M _{ϖ n (x n)}) is also T-fuzzy JU-subalgebra of X.

Proof.

Let x = (x₁, x₂, …, x _n) and y = (y₁, y₂, …, y _n) be any elements of X = $\underset{j=1}{\overset{n}{\pi }}$ X _j. Then

Definition 11.

Let ϖ₁ and ϖ₂ be fuzzy sets in X and T be a t-norm. Then the T-fuzzy product of ϖ₁ and ϖ₂ denoted by _{[ϖ1 · ϖ2]}, is defined by M_{[ϖ1 · ϖ2]} (x) = T (M _ϖ1 (x), M _ϖ2 (x))∀x ∈ X .

Also _{[ϖ1 · ϖ2]} = $M_{[{\varpi }_2·{\varpi }_1]}$

Theorem 19.

Let ϖ₁ and ϖ₂ be T-fuzzy JU-subalgebras of X. If T′ is a T-norm which dominates T, i.e. T′ (T (p, q), T (r, s)) ≥ T (T′ (p, r), T′ (q, s)), ∀p, q, r, s ∈ [0, 1]. Then the T′ fuzzy product of ϖ₁ and ϖ₂, [ϖ₁ . ϖ2]′ is a T-fuzzy JU-subalgebra of X.

Proof.

Let x, y ∈ X, then

Theorem 20.

The Cartesian product of any two T-fuzzy JU-ideals of X is also a T-fuzzy JU-ideal of X.

Proof.

Suppose ϖ₁ = (X, M _ϖ1) and ϖ₂ = (Y, M _ϖ2) be two T-fuzzy JU-ideals of a JU-algebra X. Then let x, y ∈ X × Y. Now

And, let (x₁, x₂), (y₁, y₂) ∈ X × Y. Then