diff --git a/src/ast/rewriter/finite_set_rewriter.cpp b/src/ast/rewriter/finite_set_rewriter.cpp
index 03edbfb68..3773a6483 100644
--- a/src/ast/rewriter/finite_set_rewriter.cpp
+++ b/src/ast/rewriter/finite_set_rewriter.cpp
@@ -31,55 +31,169 @@ br_status finite_set_rewriter::mk_app_core(func_decl * f, unsigned num_args, exp
     case OP_FINITE_SET_SUBSET:
         SASSERT(num_args == 2);
         return mk_subset(args[0], args[1], result);
+    case OP_FINITE_SET_SINGLETON:
+        SASSERT(num_args == 1);
+        return mk_singleton(args[0], result);
+    case OP_FINITE_SET_IN:
+        SASSERT(num_args == 2);
+        return mk_in(args[0], args[1], result);
     default:
         return BR_FAILED;
     }
 }
 
 br_status finite_set_rewriter::mk_union(unsigned num_args, expr * const * args, expr_ref & result) {
-    // Handle binary case - check if both arguments are the same
-    // set.union(x, x) -> x
+    // Idempotency: set.union(x, x) -> x
     if (num_args == 2 && args[0] == args[1]) {
         result = args[0];
         return BR_DONE;
     }
     
-    // Additional simplifications can be added here
-    // For example: set.union(x, empty) -> x
-    // But for now, we keep it minimal as per requirements
+    // Identity: set.union(x, empty) -> x or set.union(empty, x) -> x
+    if (num_args == 2) {
+        if (m_util.is_empty(args[0])) {
+            result = args[1];
+            return BR_DONE;
+        }
+        if (m_util.is_empty(args[1])) {
+            result = args[0];
+            return BR_DONE;
+        }
+        
+        // Absorption: set.union(x, set.intersect(x, y)) -> x
+        expr* a1, *a2;
+        if (m_util.is_intersect(args[1], a1, a2)) {
+            if (args[0] == a1 || args[0] == a2) {
+                result = args[0];
+                return BR_DONE;
+            }
+        }
+        
+        // Absorption: set.union(set.intersect(x, y), x) -> x
+        if (m_util.is_intersect(args[0], a1, a2)) {
+            if (args[1] == a1 || args[1] == a2) {
+                result = args[1];
+                return BR_DONE;
+            }
+        }
+    }
     
     return BR_FAILED;
 }
 
 br_status finite_set_rewriter::mk_intersect(unsigned num_args, expr * const * args, expr_ref & result) {
-    // set.intersect(x, x) -> x
+    // Idempotency: set.intersect(x, x) -> x
     if (num_args == 2 && args[0] == args[1]) {
         result = args[0];
         return BR_DONE;
     }
     
+    // Annihilation: set.intersect(x, empty) -> empty or set.intersect(empty, x) -> empty
+    if (num_args == 2) {
+        if (m_util.is_empty(args[0])) {
+            result = args[0];
+            return BR_DONE;
+        }
+        if (m_util.is_empty(args[1])) {
+            result = args[1];
+            return BR_DONE;
+        }
+        
+        // Absorption: set.intersect(x, set.union(x, y)) -> x
+        expr* a1, *a2;
+        if (m_util.is_union(args[1], a1, a2)) {
+            if (args[0] == a1 || args[0] == a2) {
+                result = args[0];
+                return BR_DONE;
+            }
+        }
+        
+        // Absorption: set.intersect(set.union(x, y), x) -> x
+        if (m_util.is_union(args[0], a1, a2)) {
+            if (args[1] == a1 || args[1] == a2) {
+                result = args[1];
+                return BR_DONE;
+            }
+        }
+    }
+    
     return BR_FAILED;
 }
 
 br_status finite_set_rewriter::mk_difference(expr * arg1, expr * arg2, expr_ref & result) {
     // set.difference(x, x) -> set.empty
     if (arg1 == arg2) {
-        // Get the set sort directly from the argument
         sort* set_sort = arg1->get_sort();
         SASSERT(m_util.is_finite_set(set_sort));
-        
-        // Call mk_empty with set_sort directly as suggested
         result = m_util.mk_empty(set_sort);
         return BR_DONE;
     }
     
+    // Identity: set.difference(x, empty) -> x
+    if (m_util.is_empty(arg2)) {
+        result = arg1;
+        return BR_DONE;
+    }
+    
+    // Annihilation: set.difference(empty, x) -> empty
+    if (m_util.is_empty(arg1)) {
+        result = arg1;
+        return BR_DONE;
+    }
+    
     return BR_FAILED;
 }
 
 br_status finite_set_rewriter::mk_subset(expr * arg1, expr * arg2, expr_ref & result) {
-    // set.subset(x, y) -> set.intersect(x, y) = x
+    // set.subset(x, x) -> true
+    if (arg1 == arg2) {
+        result = m().mk_true();
+        return BR_DONE;
+    }
+    
+    // set.subset(empty, x) -> true
+    if (m_util.is_empty(arg1)) {
+        result = m().mk_true();
+        return BR_DONE;
+    }
+    
+    // set.subset(x, empty) -> x = empty
+    if (m_util.is_empty(arg2)) {
+        result = m().mk_eq(arg1, arg2);
+        return BR_REWRITE1;
+    }
+    
+    // General case: set.subset(x, y) -> set.intersect(x, y) = x
     expr_ref intersect(m());
     intersect = m_util.mk_intersect(arg1, arg2);
     result = m().mk_eq(intersect, arg1);
     return BR_REWRITE3;
 }
+
+br_status finite_set_rewriter::mk_singleton(expr * arg, expr_ref & result) {
+    // Singleton is already in normal form, no simplifications
+    return BR_FAILED;
+}
+
+br_status finite_set_rewriter::mk_in(expr * elem, expr * set, expr_ref & result) {
+    // set.in(x, empty) -> false
+    if (m_util.is_empty(set)) {
+        result = m().mk_false();
+        return BR_DONE;
+    }
+    
+    // set.in(x, singleton(y)) checks
+    expr* singleton_elem;
+    if (m_util.is_singleton(set, singleton_elem)) {
+        // set.in(x, singleton(x)) -> true (when x is the same)
+        if (elem == singleton_elem) {
+            result = m().mk_true();
+            return BR_DONE;
+        }
+        // set.in(x, singleton(y)) -> x = y (when x != y)
+        result = m().mk_eq(elem, singleton_elem);
+        return BR_REWRITE1;
+    }
+    
+    return BR_FAILED;
+}
diff --git a/src/test/finite_set_rewriter.cpp b/src/test/finite_set_rewriter.cpp
index 3aaf9ec83..13e9c4091 100644
--- a/src/test/finite_set_rewriter.cpp
+++ b/src/test/finite_set_rewriter.cpp
@@ -36,9 +36,9 @@ static void test_union_idempotent() {
     app_ref s1(fsets.mk_range(zero, ten), m);
     
     // Test set.union(s1, s1) -> s1
-    expr* args[2] = { s1, s1 };
+    app_ref union_app(fsets.mk_union(s1, s1), m);
     expr_ref result(m);
-    br_status st = rw.mk_union(2, args, result);
+    br_status st = rw.mk_app_core(union_app->get_decl(), union_app->get_num_args(), union_app->get_args(), result);
     
     ENSURE(st == BR_DONE);
     ENSURE(result == s1);
@@ -59,9 +59,9 @@ static void test_intersect_idempotent() {
     app_ref s1(fsets.mk_range(zero, ten), m);
     
     // Test set.intersect(s1, s1) -> s1
-    expr* args[2] = { s1, s1 };
+    app_ref intersect_app(fsets.mk_intersect(s1, s1), m);
     expr_ref result(m);
-    br_status st = rw.mk_intersect(2, args, result);
+    br_status st = rw.mk_app_core(intersect_app->get_decl(), intersect_app->get_num_args(), intersect_app->get_args(), result);
     
     ENSURE(st == BR_DONE);
     ENSURE(result == s1);
@@ -82,8 +82,9 @@ static void test_difference_same() {
     app_ref s1(fsets.mk_range(zero, ten), m);
     
     // Test set.difference(s1, s1) -> empty
+    app_ref diff_app(fsets.mk_difference(s1, s1), m);
     expr_ref result(m);
-    br_status st = rw.mk_difference(s1, s1, result);
+    br_status st = rw.mk_app_core(diff_app->get_decl(), diff_app->get_num_args(), diff_app->get_args(), result);
     
     ENSURE(st == BR_DONE);
     ENSURE(fsets.is_empty(result));
@@ -106,8 +107,9 @@ static void test_subset_rewrite() {
     app_ref s2(fsets.mk_range(zero, twenty), m);
     
     // Test set.subset(s1, s2) -> set.intersect(s1, s2) = s1
+    app_ref subset_app(fsets.mk_subset(s1, s2), m);
     expr_ref result(m);
-    br_status st = rw.mk_subset(s1, s2, result);
+    br_status st = rw.mk_app_core(subset_app->get_decl(), subset_app->get_num_args(), subset_app->get_args(), result);
     
     ENSURE(st == BR_REWRITE3);
     ENSURE(m.is_eq(result));
@@ -148,10 +150,187 @@ static void test_mk_app_core() {
     ENSURE(result == s1);
 }
 
+static void test_union_with_empty() {
+    ast_manager m;
+    reg_decl_plugins(m);
+    
+    finite_set_util fsets(m);
+    finite_set_rewriter rw(m);
+    arith_util arith(m);
+    
+    // Create a set and empty set
+    sort_ref int_sort(arith.mk_int(), m);
+    expr_ref zero(arith.mk_int(0), m);
+    expr_ref ten(arith.mk_int(10), m);
+    app_ref s1(fsets.mk_range(zero, ten), m);
+    app_ref empty_set(fsets.mk_empty(s1->get_sort()), m);
+    
+    // Test set.union(s1, empty) -> s1
+    app_ref union_app1(fsets.mk_union(s1, empty_set), m);
+    expr_ref result1(m);
+    br_status st1 = rw.mk_app_core(union_app1->get_decl(), union_app1->get_num_args(), union_app1->get_args(), result1);
+    ENSURE(st1 == BR_DONE);
+    ENSURE(result1 == s1);
+    
+    // Test set.union(empty, s1) -> s1
+    app_ref union_app2(fsets.mk_union(empty_set, s1), m);
+    expr_ref result2(m);
+    br_status st2 = rw.mk_app_core(union_app2->get_decl(), union_app2->get_num_args(), union_app2->get_args(), result2);
+    ENSURE(st2 == BR_DONE);
+    ENSURE(result2 == s1);
+}
+
+static void test_intersect_with_empty() {
+    ast_manager m;
+    reg_decl_plugins(m);
+    
+    finite_set_util fsets(m);
+    finite_set_rewriter rw(m);
+    arith_util arith(m);
+    
+    // Create a set and empty set
+    sort_ref int_sort(arith.mk_int(), m);
+    expr_ref zero(arith.mk_int(0), m);
+    expr_ref ten(arith.mk_int(10), m);
+    app_ref s1(fsets.mk_range(zero, ten), m);
+    app_ref empty_set(fsets.mk_empty(s1->get_sort()), m);
+    
+    // Test set.intersect(s1, empty) -> empty
+    app_ref intersect_app1(fsets.mk_intersect(s1, empty_set), m);
+    expr_ref result1(m);
+    br_status st1 = rw.mk_app_core(intersect_app1->get_decl(), intersect_app1->get_num_args(), intersect_app1->get_args(), result1);
+    ENSURE(st1 == BR_DONE);
+    ENSURE(result1 == empty_set);
+    
+    // Test set.intersect(empty, s1) -> empty
+    app_ref intersect_app2(fsets.mk_intersect(empty_set, s1), m);
+    expr_ref result2(m);
+    br_status st2 = rw.mk_app_core(intersect_app2->get_decl(), intersect_app2->get_num_args(), intersect_app2->get_args(), result2);
+    ENSURE(st2 == BR_DONE);
+    ENSURE(result2 == empty_set);
+}
+
+static void test_difference_with_empty() {
+    ast_manager m;
+    reg_decl_plugins(m);
+    
+    finite_set_util fsets(m);
+    finite_set_rewriter rw(m);
+    arith_util arith(m);
+    
+    // Create a set and empty set
+    sort_ref int_sort(arith.mk_int(), m);
+    expr_ref zero(arith.mk_int(0), m);
+    expr_ref ten(arith.mk_int(10), m);
+    app_ref s1(fsets.mk_range(zero, ten), m);
+    app_ref empty_set(fsets.mk_empty(s1->get_sort()), m);
+    
+    // Test set.difference(s1, empty) -> s1
+    app_ref diff_app1(fsets.mk_difference(s1, empty_set), m);
+    expr_ref result1(m);
+    br_status st1 = rw.mk_app_core(diff_app1->get_decl(), diff_app1->get_num_args(), diff_app1->get_args(), result1);
+    ENSURE(st1 == BR_DONE);
+    ENSURE(result1 == s1);
+    
+    // Test set.difference(empty, s1) -> empty
+    app_ref diff_app2(fsets.mk_difference(empty_set, s1), m);
+    expr_ref result2(m);
+    br_status st2 = rw.mk_app_core(diff_app2->get_decl(), diff_app2->get_num_args(), diff_app2->get_args(), result2);
+    ENSURE(st2 == BR_DONE);
+    ENSURE(result2 == empty_set);
+}
+
+static void test_subset_with_empty() {
+    ast_manager m;
+    reg_decl_plugins(m);
+    
+    finite_set_util fsets(m);
+    finite_set_rewriter rw(m);
+    arith_util arith(m);
+    
+    // Create a set and empty set
+    sort_ref int_sort(arith.mk_int(), m);
+    expr_ref zero(arith.mk_int(0), m);
+    expr_ref ten(arith.mk_int(10), m);
+    app_ref s1(fsets.mk_range(zero, ten), m);
+    app_ref empty_set(fsets.mk_empty(s1->get_sort()), m);
+    
+    // Test set.subset(empty, s1) -> true
+    app_ref subset_app1(fsets.mk_subset(empty_set, s1), m);
+    expr_ref result1(m);
+    br_status st1 = rw.mk_app_core(subset_app1->get_decl(), subset_app1->get_num_args(), subset_app1->get_args(), result1);
+    ENSURE(st1 == BR_DONE);
+    ENSURE(m.is_true(result1));
+    
+    // Test set.subset(s1, s1) -> true
+    app_ref subset_app2(fsets.mk_subset(s1, s1), m);
+    expr_ref result2(m);
+    br_status st2 = rw.mk_app_core(subset_app2->get_decl(), subset_app2->get_num_args(), subset_app2->get_args(), result2);
+    ENSURE(st2 == BR_DONE);
+    ENSURE(m.is_true(result2));
+}
+
+static void test_in_singleton() {
+    ast_manager m;
+    reg_decl_plugins(m);
+    
+    finite_set_util fsets(m);
+    finite_set_rewriter rw(m);
+    arith_util arith(m);
+    
+    // Create elements and singleton
+    expr_ref five(arith.mk_int(5), m);
+    expr_ref ten(arith.mk_int(10), m);
+    app_ref singleton_five(fsets.mk_singleton(five), m);
+    
+    // Test set.in(five, singleton(five)) -> true
+    app_ref in_app1(fsets.mk_in(five, singleton_five), m);
+    expr_ref result1(m);
+    br_status st1 = rw.mk_app_core(in_app1->get_decl(), in_app1->get_num_args(), in_app1->get_args(), result1);
+    ENSURE(st1 == BR_DONE);
+    ENSURE(m.is_true(result1));
+    
+    // Test set.in(ten, singleton(five)) -> ten = five
+    app_ref in_app2(fsets.mk_in(ten, singleton_five), m);
+    expr_ref result2(m);
+    br_status st2 = rw.mk_app_core(in_app2->get_decl(), in_app2->get_num_args(), in_app2->get_args(), result2);
+    ENSURE(st2 == BR_REWRITE1);
+    ENSURE(m.is_eq(result2));
+}
+
+static void test_in_empty() {
+    ast_manager m;
+    reg_decl_plugins(m);
+    
+    finite_set_util fsets(m);
+    finite_set_rewriter rw(m);
+    arith_util arith(m);
+    
+    // Create element and empty set
+    sort_ref int_sort(arith.mk_int(), m);
+    expr_ref five(arith.mk_int(5), m);
+    parameter param(int_sort.get());
+    sort_ref set_sort(m.mk_sort(fsets.get_family_id(), FINITE_SET_SORT, 1, &param), m);
+    app_ref empty_set(fsets.mk_empty(set_sort), m);
+    
+    // Test set.in(five, empty) -> false
+    app_ref in_app(fsets.mk_in(five, empty_set), m);
+    expr_ref result(m);
+    br_status st = rw.mk_app_core(in_app->get_decl(), in_app->get_num_args(), in_app->get_args(), result);
+    ENSURE(st == BR_DONE);
+    ENSURE(m.is_false(result));
+}
+
 void tst_finite_set_rewriter() {
     test_union_idempotent();
     test_intersect_idempotent();
     test_difference_same();
     test_subset_rewrite();
     test_mk_app_core();
+    test_union_with_empty();
+    test_intersect_with_empty();
+    test_difference_with_empty();
+    test_subset_with_empty();
+    test_in_singleton();
+    test_in_empty();
 }